# A Model-Based Investigation of the Performance Robustness of the Deflector Jet Servo Valve
## 3 Methodology of Modeling Hydraulic Amplifier
### Deflector Jet and Pressure Recovery in Receivers  
For the deflector jet, the total momentum is  
\[M^{\prime}=4\rho u_{0}^{\prime}{}^{2}b_{0}^{\prime}z_{e} \tag{17}\]  
However, not all fluid flowing out of the deflector is used to generate the recovery pressure of the receivers, as depicted in Figure 6. In the jet direction, the momentum acting on the shunt wedge and the two receivers can be estimated as  
\[M_{e}^{\prime}=4\rho u_{0}^{\prime}{}^{2}b_{0}^{\prime}l_{c}^{\prime} \tag{18}\]  
Simulation results [24] show that each receiver has a fluid surface to sustain the jet impact, which is referred to as the effective impact area. Approximately, the pressure in this area is uniform and equal to the pressure in the receiver. Figure 6: Solid impact surface and effective impact area. **Assumption 7**. : _The effective impact area is an isobaric surface with the pressure equal to the internal pressure of the receiver. The two receivers have the same effective impact area, which is fixed and unaffected by the deflectormovement._  
In view of the fact that the momentum in the jet direction is zero after impact, the momentum equation in this direction can be written as  
\[4\rho b_{0}^{\prime}l_{c}^{\prime}{u^{\prime}}_{0}^{2}=\iint\limits_{S_{m}}p \leavevmode\
obreak\ dydz+(p_{1}+p_{2})S_{r} \tag{19}\]  
where \(S_{m}\) is the solid impact surface on the shunt wedge; \(p\) denotes the pressure distribution on \(S_{m}\); \(p_{1}\) and \(p_{2}\) are the receiver pressures; and \(S_{r}\) is the effective impact area of the receiver, decided by the effective impact width \(l_{e}\). A visual description is provided in Figure 6.  
Consider the specific situation of the receiver. After impacting on the shunt wedge, the fluid flowing through the receiver obtains a large velocity component parallel to the impact surface, resulting in emergence of a low-pressure region adhering to the side of the shunt wedge, as illustrated in Figure 7. After flowing past the low-pressure region, the fluid continues to be deflected by the pressure in the receiver. Meanwhile, this deflected flow gives the closed chamber of the receiver a continuous reaction to drive the sliding spool. For a deflector jet valve, this process reflects the formation mechanism of the working pressure. As the flow continues to deflect, the momentum in the jet direction gradually decreases to zero, and then, the flow is no longer able to provide effective momentum for the receiver. Consequently, a smooth boundary is designed to block the flow and generate a high-pressure region, which encloses the chamber of the receiver. Thus, a low-pressure region, a high-pressure region, and a working pressure region constitute the entire pressure distribution near the receiver. Only the distance between the high-pressure region and the low-pressure region is the effective impact width \(l_{e}\) (Figure 7). This high-pressure region can be supposed to originate from another jet impact, which comes into being just outside the shunt wedge and the low-pressure region. Experimental data [30] show that when a jet travels past half of the total distance between the jet origin and the impact wall, the pressure along the jet axis will rise significantly, meaning that the jet reaches the high-pressure region. It follows that the effective impact length can be calculated as  
\[l_{e}\leavevmode\
obreak\ =l_{r}-\frac{1}{2}(l_{r}-l_{l})-l_{l} \tag{20}\]  
where \(l_{l}\) is the width of the low-pressure region and \(l_{r}\) is the total width of the receiver opening. With reference to the numerical simulation results (Figure 7), the width of the low-pressure region is approximately constant, with a value 0.02 mm. Then, it is learned from Equation (20) that \(l_{e}\) = 0.16 mm, and \(S_{r}\) has an area of 0.085 mm\({}^{2}\). Consider the situation of the shunt wedge. Assume that the pressure on the solid impact surface conforms to the Gaussian distribution as follows.