## 3 Modeling and Simulation  
The improved spool is shown in Figure 2. In this paper, the diameter of the right spool stem is reduced on the basis of the original spool, so that the spool stem is in the shape of a round platform. A triangular groove is circumscribed on the right step, which is composed of surface 3 and surface 4. After part of the liquid flow is diverted through surface 3 and surface 4, the flow direction changes, that is, a leftward flow component is generated. This part of the liquid flow reversely impacts the liquid flow flowing to the right, which increases the jet angle at the outlet and reduces the flow force. As the diameter on the right side of the spool stem is reduced to form a round platform, the space on the right side of the spool stem is larger. This allows more fluid to flow through surface 3 and surface 4, further reducing the flow force. As shown in Figure 3, the main structural parameters of the improved spool are the right spool stem's end diameter, \(D\), the triangular groove's bottom diameter, \(D_{1}\), and the groove depth, \(H\).  
The three-dimensional model was established in SolidWorks and imported into ANSYS/Fluent to simulate the fluid domain. In this paper, the initial parameters are \(D\) = 8 mm, \(H\) = 4 mm, \(D_{1}\) = 16 mm. The diameter of the unmodified stem is 12 mm. Reynolds number is 2351. There are three turbulence models K-\(\epsilon\), K-\(\omega\), and Reynolds stress in ANSYS/Fluent. K-\(\epsilon\) model is the most widely used turbulence model in the research of hydraulic valve. Based on this model, the valve is simulated and analyzed in this paper. Turbulent kinetic energy, K, and dissipation, \(\epsilon\), can be derived from the migration equation, as follows:  
\[\frac{\partial}{\partial t}(\rho k)+\frac{\partial}{\partial x_{i}}(\rho ku_{ i})=\frac{\partial}{\partial x_{j}}[(\mu+\frac{\mu_{t}}{\sigma_{k}})\frac{ \partial k}{\partial x_{j}}]+G_{k}+G_{b}-\rho\epsilon-Y_{M}+S_{k} \tag{12}\]  
\[\frac{\partial}{\partial t}(\rho\epsilon)+\frac{\partial}{\partial x_{i}}( \rho\epsilon u_{i})=\frac{\partial}{\partial x_{j}}[(\mu+\frac{\mu_{t}}{ \sigma_{\epsilon}})\frac{\partial_{\epsilon}}{\partial x_{j}}]+C_{1\epsilon }\frac{\epsilon}{k}(G_{k}+C_{3\epsilon}G_{b})-C_{2\epsilon}\rho\frac{\epsilon ^{2}}{k}+S_{\epsilon} \tag{13}\]  
where \(G_{k}\) is the turbulent kinetic energy generated by the average velocity gradient. \(G_{b}\) is the turbulent kinetic energy generated by liquid buoyancy. \(Y_{M}\) is the fluctuation expansion of the overall dissipation rate in the compressible turbulent flow. \(C_{1\epsilon}\), \(C_{2\epsilon}\) and \(C_{3\epsilon}\) are model constants. \(\sigma_{k}\) and \(\sigma_{\epsilon}\) are the turbulent Prandtl number of \(k\) and \(\epsilon\), \(S_{k}\) and \(S_{\epsilon}\) are user-defined values. \(\mu_{t}\) is turbulent viscosity, which can be expressed as follows:  
\[\mu_{t}=\rho C_{\mu}\frac{k^{2}}{\epsilon} \tag{14}\]  
In this paper, the form of the pressure inlet and pressure outlet are adopted; the inlet pressure is 3 MPa and the outlet pressure is 0.5 MPa, and the fit clearance between the spool and the spool sleeve is ignored. The fluid density is 870 kg \(\cdot\) m\({}^{-3}\). The dynamic viscosity is 0.04 kg \(\cdot\) m\({}^{-1}\cdot\) s\({}^{-1}\).