## 4 Results
### Shape of the Cavity
#### 4.3.1 The Influence of Shape of the Cavity in Cavity-Single Tube Flow  
Figure 18 illustrates the effect of the variation of \(L/D\) on the variation trajectory of \(\overline{m}\) in the cavity-tube flow path. During the process of \(L/D\) increasing from 0.15 to 0.40, the amplitude and phase of the variation trajectory of \(\overline{m}\) change slightly; however, when \(L/D\) increases from 0.40 to 1.86, the variation trajectory of \(\overline{m}\) changes significantly. An analysis of the influence patterns at different stages is conducted.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{**Cavity-Single Tube Flow**} & \multicolumn{2}{c}{**Cavity-Double Tube Flow**} \\ \hline \(l_{1}/d\) & 0.15 & 0.23 & 0.40 & 0.78 & 1.86 \\ \(l_{2}/d\) & 0 & & 1.5 & \\ \(t_{char}/\mathbf{s}\) & 2.24 & & 2.67 & \\ \hline \hline \end{tabular}
\end{table}
Table 11: Comparative analysis of the effects of \(L/D\) on operating conditions.  
Figure 18: (a) Evolution of \(\overline{m}\); (b) Comparison of oscillation amplitude of \(\overline{m}\); (c) Comparison of period of \(T\).  
Select \(t_{s1}\) and \(t_{s2}\) as the comparison moments for the venting and inflation processes, respectively, and compare the variation in the distribution range of \(Ma\) within the cavity during these processes, as shown in Figure 19. The variation in \(L/D\) is achieved by altering the axial and radial distances of the cavity, thereby influencing the fast transient response of the flow path through the constraints of the CIDZ.  
Figure 19: Comparison of flow field \(Ma\) variation in different \(L/D\) cavity-single tube flow path (**a**) discharge process; (**b**) charge process.  
During the venting process, as \(L/D\) increases, the axial distance constraint gradually weakens, while the radial distance constraint gradually strengthens. Prior to the axial distance becoming unconstrained, CIDZ increases slowly; however, once the axial distance is unconstrained, CIDZ increases significantly. Due to differences in flow direction, variations in the axial distance of the cavity exert a greater influence on the inflation process. Similarly, during the inflation process, as \(L/D\) increases, the axial distance constraint weakens, allowing the flow to fully develop within the cavity, which results in an increase in CIDZ. When the radial distance is constrained, the flow develops along the axis, leading to an increase in the distribution of inertial forces within the cavity.  
Figure 20 illustrates the impact of various parameters \(L/D\) on the axial load response during the fast transient response process of the cavity-tube flow path. Within the range of 0.15 to 0.4, the effects of \(L/D\) on \(\overline{F_{A}}\) and \(\overline{F_{B}}\) are minimal, remaining above 2%. As \(L/D\) increases from 0.4 to 1.86, the variations in \(\overline{F_{A}}\) and \(\overline{F_{B}}\) increase significantly, reaching a maximum of 7.5%. Changes in \(L/D\) directly influence the phase difference between the response processes of \(\overline{F_{A}}\) and \(\overline{F_{B}}\); consequently, \(\Delta\overline{F}\) continuously varies with \(L/D\). During the increase from 0.15 to 1.86, \(\left(\Delta\overline{F}\right)_{\text{max}}\) rises by more than 250%, while \(\left(\Delta\overline{F}\right)_{\text{min}}\) decreases by over 400%. Specific parameters are detailed in Table 12.  
Overall, when \(L/D\) is small, it does not restrict the CIDZ range and thus has minimal impact on the fast transient response process of the downstream flow path at the sudden change boundary. It also does not affect the axial load response process within the cavity. However, as \(L/D\) increases, it gradually begins to restrict the CIDZ range, thereby influencing the inertial forces within the cavity. Considering the effect of the increase in \(L/D\) on the phase difference in the axial load response history at both ends of the cavity, variations in \(L/D\) complicate the evolution of the axial load in the flow path.