# A Model-Based Investigation of the Performance Robustness of the Deflector Jet Servo Valve
## 3 Methodology of Modeling Hydraulic Amplifier
### Initial Jet
Based on Equation (4) and a linear assumption, the outline in an arbitrary perpendicular section \(D_{h}\) at the location \(x\) can be determined by  
\[y_{h}=-\frac{b_{0}}{L_{0}}\cdot x+b_{0} \tag{5}\]  
\[z_{h}=-\frac{l_{0}}{L_{0}}\cdot x+l_{0} \tag{6}\]  
For the critical section, \(y_{h}=y_{thr}\) and \(z_{h}=z_{thr}\), as depicted in Figure 4.  
**Assumption 4**. : _The flow velocity outside the potential core conforms to the Gaussian distribution, and its eigen half-width is approximately proportional to the orifice dimension in each direction._  
Hence, the outside flow velocity in the section \(D_{h}\) can be calculated by  
\[\frac{u}{u_{0}}=\exp\left[-\frac{(y-y_{h})^{2}+\beta^{2}(z-z_{h})^{2}}{b_{e} ^{2}}\right] \tag{7}\]  
in which \(u\) is the velocity at the coordinates \((y,z)\), \(\beta=\frac{b_{0}}{l_{0}}\) is the dimension factor, and \(b_{e}\) is the eigen half-width of the entrainment, with  
\[b_{e}=ex \tag{8}\]  
where \(\varepsilon\) is the expansion coefficient of the theoretical eigen boundary of the jet, as illustrated in Figure 3.  
Figure 3: Early-terminated initial jet in deflector. For a perpendicular section \(D_{h}\), the momentum flux (momentum for short) can be calculated as  
\[M=\iint\limits_{D_{h}}\rho u^{2}dy\;dz \tag{9}\]  
Additionally, the initial momentum at the orifice is  
\[M_{0}=4\rho u_{0}^{2}b_{0}l_{0} \tag{10}\]  
The conservation of momentum of the initial jet can be represented as  
\[y_{h}=-\frac{b_{0}}{L_{0}}\cdot x+b_{0} \tag{5}\]0  
According to Equations (5)-(8) and (11), we can obtain a different \(\varepsilon\) for each \(x\), which means a nonlinear expansion of the theoretical eigen boundary. Through data fitting, the coordinate dependence of \(\varepsilon\) can be approximated by Equation (12) and \[y_{h}=-\frac{b_{0}}{L_{0}}\cdot x+b_{0} \tag{5}\]1  
\[y_{h}=-\frac{b_{0}}{L_{0}}\cdot x+b_{0} \tag{5}\]2 \begin{table}
\begin{tabular}{c c c c} \hline \hline
**N** & \(a_{n}\) & \(b_{n}\) & \(c_{n}\) \\ \hline
1 & \(1.74\times 10^{-1}\) & \(5.325\times 10^{-5}\) & \(1.266\times 10^{-4}\) \\
2 & \(2.536\times 10^{-2}\) & \(2.144\times 10^{-4}\) & \(7.328\times 10^{-5}\) \\
3 & \(2.345\times 10^{-2}\) & \(8.253\times 10^{-5}\) & \(8.253\times 10^{-5}\) \\
4 & \(8.541\times 10^{-3}\) & \(4.204\times 10^{-4}\) & \(8.633\times 10^{-5}\) \\
5 & \(1.371\times 10^{-1}\) & \(-3.103\times 10^{-4}\) & \(2.532\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Coefficients in Equation (12). Figure 4: Critical section of deflector action. Hence, with Equations (5), (6), (8) and (12) substituted into Equation (7), the velocity distribution outside the potential core can be described by  
\[u=u_{0}\exp\left\{-\frac{\left(y+\frac{b_{0}}{L_{0}}\cdot x-b_{0}\right)^{2}+ \left[\beta\left(z+\frac{l_{0}}{L_{0}}\cdot x-l_{0}\right)\right]^{2}}{\left[ \sum_{1}^{5}a_{n}\exp\left[-\left(\frac{x-b_{0}}{c_{n}}\right)^{2}\right]\cdot x \right]^{2}}\right\} \tag{13}\]  
Thus, the entire velocity distribution of the initial jet can be characterized as Equation (13) and the uniform velocity \(u_{0}\) in the potential core. In the critical section, the velocity distribution will be the interface condition for the pressure recovery region.