## 4 Results and Discussion
### Oscillatory Flow Frequency Effects Study  
Figure 8 shows the relationship between the linear resistance \(R_{\text{LA}}\) and oscillatory flow velocity amplitude \(M_{\text{v}}\) of the communication port at different oscillatory flow frequencies. The impact of oscillatory flow frequency on linear resistance is relatively small when it is below about 20 Hz. This feature is beneficial for quasi-steady assumption. However, when the frequency exceeds about 20 Hz, the correlation between linear resistance and  
Figure 7: Linear resistance under different pressure amplitudes. velocity amplitude appears to shift. The communication port appears to have lost its asymmetry and the losses caused by injection have weakened, leading to a linear increase in its resistance with velocity amplitude. Figure 9 provides velocity vector plots at oscillatory flow frequencies of 1 Hz, 10 Hz, and 50 Hz under the velocity amplitude of 1.2 m/s. When the oscillatory flow frequency is below 20 Hz (taking 1 Hz and 10 Hz condition as examples), a periodically moving vortex forms in the branch. According to boundary conditions, particles of the fluid inside the branch pipe undergo reciprocating motion, the trajectory of which can be transformed into a vortex affected by the asymmetric flow through the communication port or be maintained along the pipeline direction if not affected. Keeping the amplitude of oscillatory flow constant, the influenced region of  
Figure 8: Linear resistance under different oscillatory flow frequencies. Figure 9: Streamline diagram (\(\widetilde{W}_{\rm A}=1.2\) m/s). asymmetric flow should decrease with the increase in oscillation frequency and would appear to form a stable vortex when viewed from streamline at any moment. The increase in the oscillatory flow velocity amplitude will enhance the stability of communication port flow patterns, but this phenomenon cannot be avoided. Figure 10 provides velocity vector plots at oscillatory flow frequencies of 1 Hz, 10 Hz, and 50 Hz under the flow velocity amplitude of 30 m/s. As the flow rate ratio decreases, the effect of vertical supply pipeline flow diminishes, enhancing the symmetry of the communication port. Consequently, the deviation from the orifice flow model gradually decreases. A correlation likely exists between changes in flow patterns and the attenuation of linear resistance. Focusing on the shape of the linear resistance curve, it becomes evident that at higher oscillatory flow frequencies, the intercept characterizing the asymmetry of communication port flow shortens, meaning that the linear resistance and oscillatory flow velocity amplitude tend to exhibit a more linear correlation. And the streamline diagram provides a plausible explanation: the stable vortex generated by high-frequency flow results in symmetrical flow velocity on both sides of the communication port. As shown in Figure 11, the presence of vortices leads to vertical flow on the right side of the communication port, to a certain extent resulting in the symmetry in the flow on both sides of the communication port. In the quasi-steady assumption-based linear resistance calculation method of the communication port flow model, the asymmetry of the flow on two sides plays a crucial role in determining its linear resistance variation, which is the cause of the injection phenomenon and the variation in communication port local loss. However, under high-oscillation-frequency conditions, the presence of vortices alters this situation, causing the linear resistance curve concerning oscillation amplitude to pass through the origin. On the other hand, it can be observed that as the oscillatory flow frequency increases, the volume of vortices decreases, consequently leading to a reduction in local loss, which suggests that the flow coefficient of the communication port will decrease with increasing frequency. Figure 10: Streamline diagram (\(\widetilde{W}_{\mathrm{A}}=30\,\mathrm{m/s}\)).