## 3 Analysis of Flow Force Characteristics of Slide Valves
### Flow Force Characteristics of Two Throttling Modes at Variable Openings  
The impact of valve opening on the steady flow force in the two throttling modes was investigated, with the valve opening varying from 0.2 mm to 1.6 mm in increments of 0.2 mm. The valve opening is considered positive for inlet throttling and negative for outlet throttling. Figure 6 presents the variation curves of steady flow force and flow rate with respect to valve opening for both throttling modes. The simulation results obtained from the pressure integration and momentum theorem align closely, with a maximum deviation of approximately 10%, thereby validating the accuracy of the algorithm. The flow rate characteristics for both throttling modes exhibit a near-theoretical linear variation, indicating that the mass flow rate through the valve is not the primary reason for the significant differences in flow force between the two throttling modes. The predicted results of steady flow force from the empirical formula closely match those obtained under the outlet throttling mode, with steady flow force linearly correlated to valve opening and a maximum deviation of approximately 13%. In contrast, under the inlet throttling mode, the steady flow force exhibits a nonlinear variation with increasing valve opening, showing significant differences from the empirical formula. The analysis from the previous section indicates that, at a fixed opening, the fluid momentum at the throttling valve port remains approximately equal for both throttling modes. Therefore, it is initially determined that this discrepancy arises from the momentum of the fluid at the non-throttling valve port. To further elucidate the reasons for the flow force differences, Figure 7 illustrates the variation in steady flow force at each valve port with valve opening under the two throttling modes. As the valve opening changes, the flow forces at the throttling valve ports are approximately equal for both modes. In the outlet throttling mode, the steady flow force at the non-throttling valve port exhibits a linear increase with valve opening and is com  
Figure 5: Streamline diagrams of the flow field inside the sliding valve chamber. **Left**: inlet throttling mode; **right**: outlet throttling mode. Figure 6: Variation curves of (**a**) steady flow force and (**b**) flow rate with valve opening. paratively smaller than that at the throttling valve port. In the inlet throttling mode, the flow force at the non-throttling valve port exhibits a nonlinear increase with valve opening. Once the opening reaches a certain threshold, the flow force at the non-throttling valve port can even exceed that at the throttling valve port, resulting in a shift in the overall direction of the flow force acting on the spool, causing the valve port to transition from a tendency to close to a tendency to open. Thus, it is concluded that the fluid momentum at the non-throttling valve port significantly impacts the steady flow force of the sliding valve, particularly in the inlet throttling mode at larger openings, where the fluid momentum at the non-throttling valve cannot be neglected. Figures 8 and 9 illustrate the flow velocity clouds corresponding to different valve ports under the two throttling modes. Figure 10 presents the variation curves of the fluid jet angles and axial velocities at the two valve ports in the inlet throttling mode as a function of valve opening. The figures clearly visualize the fluid velocity and jet angles within the valve chamber. In the inlet throttling mode, the jet angle of the fluid at the throttling valve port remains around \(60^{\circ}\), showing little dependence on the valve opening. In contrast, the jet angle at the non-throttling valve port significantly decreases as the opening increases, while the axial velocity exhibits an opposite trend. Compared to the inlet throttling mode, the outlet throttling mode features fewer vortices within the chamber, resulting in a more orderly flow of the fluid, with very low velocities observed at the non-throttling valve port.