## 4 Results
### Inlet and Outlet Area on the Cavity  
In the study of the effects of inlet and outlet area on the cavity, the cavity volume \(V_{Vessel}\), initial pressure ratio \(P_{0}/P_{B}\), and cavity shape \(L/D\) are maintained at 4.92 \(\times\) 10\({}^{6}\), 1.5,  
Figure 8: (a) Cavity flow; (b) Cavity-single tube flow; (c) Cavity-double tube flow. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Influencing Factors** & **Research Parameters** & **Dimensional** \\ \hline Inlet and outlet areas of the cavity & \(A\)-Inlet and Outlet Area of the Cavity & \(A/V_{Vessel}\)\({}^{\frac{2}{3}}\) \\ \hline Length of the tube near the outlet & \(l_{1}\)-Tube length & \(l_{1}/d\) \\ \hline Length of the tube away from outlet & \(l_{2}\)-Tube length & \(l_{2}/d\) \\ \hline Shape of the cavity & \(L\)-Cavity length & \(L/D\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Dimensional analysis of research parameters. and 0.4, respectively. Table 6 presents the range of variation for \(A/V_{Vessel}\)3, and subsequent research will be conducted under these conditions. The relative change in gas quantity \(\overline{m}\) within the cavity is a crucial parameter in this study for distinguishing among the various stages of the response process, with changes in \(\overline{m}\) being the direct cause of axial load variations within the cavity. Figure 9 illustrates the effect of \(A/V_{Vessel}\)3 on the temporal changes in the relative gas quantity \(\overline{m}\), as well as the relative changes in the oscillation peak \(\overline{m}_{Crest}\), oscillation trough \(\overline{m}_{Trough}\), and oscillation period \(T\) as \(A/V_{Vessel}\)3 increases, using condition \(A/V_{Vessel}\)3 = 0.011 as the baseline for comparison. As \(A/V_{Vessel}\)3 increases, the variation of \(\overline{m}\) exhibits oscillations, with the amplitude of these oscillations gradually increasing. At the same time, the oscillation period also progressively lengthens. The significant changes in the variation of \(\overline{m}\) directly and critically affect the airflow parameters within the cavity, thereby profoundly altering the evolution of the axial load within the cavity. This process clearly demonstrates the important influence of \(A/V_{Vessel}\)3 on the inertial forces within the cavity. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(A/V_{Vessel}\)3 & 0.011 & 0.043 & 0.098 & 0.174 & 0.272 \\ \hline \(t_{char}\)/s & 45.11 & 11.28 & 5.01 & 2.82 & 1.80 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparative analysis of the effect of \(A/V_{Vessel}\)3 on operating conditions. Figure 9: (**a**) Evolution of \(\overline{m}\); (**b**) Comparison of oscillation amplitude of \(\overline{m}\); (**c**) Comparison of Period of \(T\). Figure 10 provides a more intuitive illustration of the influence of \(A/V_{Vessel}\)\({}^{\frac{2}{3}}\) on the inertial forces within the cavity. Specific moments are selected for comparison based on the data presented in Figure 9. At moment \(A/V_{Vessel}\)\({}^{\frac{2}{3}}=0.011\), the variation of \(\overline{m}\) stabilizes, facilitating the determination of \(\overline{m}_{s}=0.245\) at time \(\overline{m}=0.245\). In this study, since the cavity volume, initial pressure in the flow path, and environmental pressure remain constant, \(\overline{m}_{s}\) is consistently 0.245. The first oscillation period of the response process is examined, with moment \(\overline{m}=\overline{m}_{s}\) during the depressurization process designated as \(t_{s1}\), moment \(\overline{m}=\overline{m}_{s}\) during the pressurization process designated as \(t_{Crest}\), and moment \(\overline{m}=\overline{m}_{Crest}\) during the pressurization process designated as \(t_{Crest}\).