# A Model-Based Investigation of the Performance Robustness of the Deflector Jet Servo Valve
## 3 Methodology of Modeling Hydraulic Amplifier  
### Initial Jet  
Before the formation of the initial jet, the fluid flows through a convergent rectangular channel. Then, the initial jet velocity can be governed by the following equations [26]. \[\frac{p_{0}-p_{b}}{\rho g}=\frac{{u_{0}}^{2}}{2g}+\Delta H \tag{1}\]  
\[\Delta H=\zeta_{0}\frac{u_{0}^{2}}{2g} \tag{2}\]  
where \(u_{0}\) is the average velocity at the initial jet orifice (Figure 3), \(p_{0}\) is the supply pressure of hydraulic oil, \(p_{b}\) is the pressure outside the orifice (back pressure), \(\Delta H\) is the loss of the gradual contracting flow, and \(\zeta_{0}\) is the local loss coefficient with Equation (3) [26]. \[\zeta_{0}=\frac{\lambda}{8\tan\alpha}\left[1-\left(\frac{2b_{0}}{A_{1}}\right) ^{2}\right] \tag{3}\]  
where \(\lambda\) is the frictional coefficient of the channel, given in [27] as \(\lambda=0.055\). It follows from Equations (1)-(3) that the initial velocity \(u_{0}=202.41\) m/s. Generally, the length of the potential core region for the jet can be given according to [28] as  
\[L_{0}=2.13D_{0}R_{e}^{\,0.097} \tag{4}\]  
where \(D_{0}=\frac{4\cdot 2h_{0}\cdot 2l_{0}}{2\cdot(2h_{0})+2l_{0}}\) is the hydraulic diameter, and \(R_{e}=\frac{\rho u_{0}D_{0}}{\mu}\) is the Reynolds number. Equation (4) implies that the potential core extends far beyond the scope of the deflector, as depicted in Figure 3. However, the flow in the DJV faces a complicated situation whereby the initial jet is affected by a combination of the back pressure and the V-shaped deflector. It follows that the potential core region is terminated by recovered high pressure in the deflector. Based on an investigation into data from 3D simulation [16], some assumptions about the initial jet in the deflector are presented as follows. \begin{table}
\begin{tabular}{c c c} \hline
**Parameter** & **Symbol** & **Value** \\ \hline Initial jet half-width & \(b_{0}\) & \(7.5\times 10^{-5}\) m \\ Deflector jet half-width & \(b^{\,0}\) & \(7.0\times 10^{-5}\) m \\ Inflow width & \(A_{1}\) & \(1.15\times 10^{-3}\) m \\ Shunt wedge width & \(l_{\rm m}\) & \(1.1\times 10^{-4}\) m \\ Receiver opening width & \(l_{\rm r}\) & \(1.8\times 10^{-4}\) m \\ Jet orifice thickness & \(2l_{0}\) & \(5.4\times 10^{-4}\) m \\ Upper deflector half-thickness & \(h_{\rm u}\) & \(8.5\times 10^{-4}\) m \\ Lower deflector half-thickness & \(h_{\rm j}\) & \(7.5\times 10^{-4}\) m \\ Receiver thickness & \(2^{f}\,_{\rm c}\) & \(5.4\times 10^{-4}\) m \\ Deflector jet distance & \(H\) & \(1.95\times 10^{-4}\) m \\ Inflow angle & \(\alpha\) & \(13^{\circ}\) \\ Sidewall inclination angle & \(\theta\) & \(16.53^{\circ}\) \\ Hydraulic oil density & \(\rho\) & \(849\) kg/m\({}^{3}\) \\ Supply pressure & \(p_{0}\) & \(21\) MPa \\ Back pressure & \(p_{\rm b}\) & \(3.1\) MPa \\ Gravity acceleration & \(g\) & \(9.8\) m/s\({}^{2}\) \\ Hydraulic oil viscosity & \(\mu\) & \(0.017336\) Pa-s \\ \hline \end{tabular}
\end{table} 
Table 1: Dimensions of deflector jet structure. **Assumption 1**. : _The jet can be considered unaffected by the deflector before a certain perpendicular section, which is referred to as the critical section of the deflector action (the critical section for short)._  
**Assumption 2**. : _The critical section is located by the stagnation line on the sidewall in the deflector. (Section 3.2)._  
**Assumption 3**. : _The critical section is the interface between the initial jet and the high-pressure region in the deflector._  
With the assumptions above, the two stages, the initial jet and the pressure recovery in the deflector are clearly separated by the critical section. Then, before arriving at the high-pressure region, the potential core has a varying outline along the jet direction (the \(x\)-axis in Figure 3).