**1. Introduction**

As the core component of hydraulic control systems, the electro-hydraulic servo valve has certain advantages of high performance and high reliability. Its first appearance was to the application in fighter planes during World War II, but the single-stage open-loop structure made it difficult to meet the control requirements at that time [1]. It was not until Massachusetts institute of technology (MIT) replaced the solenoid with a high-frequency permanent magnet torque motor that the servo valve ushered in its golden period of development. In 1953, the single nozzle flapper valve was firstly invented by Moog [2]. After four years, the single nozzle structure was further improved to have double nozzles by Howard [3]. In 1962, Atchley [4] invented the jet tube servo valve. Thanks to the development of electronic technology, Vanderlaan et al. [5] made the servo motors directly drive the spool movement in 1987. In 1993, Laux [6] improved and invented the rotary direct drive servo valve. However, the direct drive servo valves are still limited by insufficient motor power, leading to the frequency response not being quick enough. With the advantages of a high power density ratio and a high frequency response, jet servo valves are still popularly applied in crucial industrial applications, including aerospace, ship engineering, high-end robots, etc. At present, research on jet servo valves are still in progress.

There are many related aspects of servo valve research. For example, on the control algorithm, Samakwong et al. [7] found that a genetic algorithm (GA) could better optimize the parameters of the PID controller and control the performance of the servo valve than the Ziegler–Nichols adjustment method. With respect to mathematical modeling, Brito et al. [8] established a Hammerstein model for aerospace servo valves, the results showing that the identified model can represent the general

non-linear behavior of servo valves. With respect to hydraulic power, Zohreh et al. [9] simulated the valve core pressure under unsteady conditions. It was found that in the two-stage flapper nozzle electro-hydraulic valve the external acceleration would change the fluid pressure leaving the nozzle and produce the same effect as the external force. Ye et al. [10] established the dynamic model of the plunger pump, simulated it by Computational Fluid Dynamics (CFD), and verified it experimentally. The results show that the vibration speed of the plunger pump on the *X*<sup>F</sup> axis is higher than that on the *Y*<sup>F</sup> axis. The excitation moment *M*CY and *M*PY on the *Y*<sup>F</sup> axis contribute greatly to the vibration of the plunger pump. On the flow field characteristic, many scholars adopted CFD approaches to obtain the complex valve flow detail. Brito et al. [8] also carried out experimental and numerical studies to determine the mechanism of cavitation in the fluid region between the flapper and nozzle by using 3D models and CFD grids. Li et al. [11] observed the cavitation phenomena in the flow field from Reynolds numbers 630–2500 with the comparison of CFD simulation. They found that the computational results were in good agreement with the experimental observations and came to the conclusion that the position of the cavitation source is shown at the tip of the nozzle inner wall, the tip of the nozzle outer wall, and the front of the flapper. Chen et al. [12] revealed the effect of oil viscosity on the transient distribution of cavitation and small-size vortices, indicating the noise accompanied by the flow resonance in the nozzle. When the pressure fluctuation near the flowmeter is large enough in the two-stage servo valve, flow acoustic resonance and screaming may occur. Qian et al. [13] researched the forward and reverse flow of Al2O3-water nanofluids in micro T45-R Tesla valves at different flow rates, temperatures, and nanoparticle volume fractions by CFD on the basis of the verified numerical model, finding that the main flow percentage was proportional to the above three factors and the flow rate has the greatest influence on the polarity of the valve. Chao et al. [14] found that the inward inclined design of cylinder ports could effectively decrease the gaseous cavitation and increase the effective output flow of cylinder by using centrifugal effects of rotating fluid, which provided a new way to optimize the performance of (Electro-Hydrostatic Actuator) EHA. Qian et al. [15] used CFD to simulate the valve core diameter, single hole/porous diameter, hole diameter, and its arrangement at the bottom of the valve core steadily and instantaneously, and found that the pressure difference between the two sides increases with the increase of the diameter of the valve core and the decrease of the aperture. Meanwhile, the opening time of the main valve also increases with the increase of the diameter of the valve core. Zhang et al. [16] proposed a damping sleeve with a throttle hole. Through experiments and numerical calculation, it was found that the designed damper sleeve had a significant effect on the pressure distribution and jet direction on the surface of the cone, which can significantly reduce the flow force and the opening time of the valve.

Recently, with the improvement of working requirements, the jet flow velocity comes to a higher level with a larger Mach number. The flow compressible effect is more and more obvious. The authors acknowledge there are rarely any studies that have paid enough attention to the high-speed compressible flow oscillations in the jet servo valves. For the present paper content organization, the CFD approach is employed with a (Large Eddy Simulation) LES turbulent model to obtain the vortex flow oscillation conditions. After determining the independence of the grid, the flow field of the fluid in the servo valve with double nozzles and flappers is analyzed under the condition of changing the inlet oil flow rate and the deflection displacement of the flappers, while the force acting on the servo valve flappers under the coupling of the main valve is also discussed.

#### **2. Flow Structure and Grid Independence Analysis**

#### *2.1. Operation Principle and Structural Parameters*

The two-stage double nozzle flapper force feedback electro-hydraulic servo valve is taken as the research object. As shown in Figure 1a, when the servo valve is in the initial position, the coil is not electrified, the flapper is located in the middle of the nozzle without deflecting, the flow force acting on the flapper is offset each other, the pressure loss caused by the variable throttle hole is the same, the pressure at both ends of the main valve core is the same, and the main valve core is not moving. When the corresponding electric signal is input, the coil generates a magnetic field, which makes the torque motor produce a magnetic moment, and drives the flapper to produce the corresponding deflection angle, thus promoting the movement of the main valve core. As shown in Figure 1b, the simplified structure of the jet location, when the current flowing through the left and right coils is different, for example of *i*<sup>1</sup> > *i*2, the electromagnetic moment produced by the left coil is larger than that of the right coil, which makes the coil rotate clockwise, and makes the flapper shift to the left, thus making the distance between the flapper of the jet flapper valve and the two nozzles different, the left side smaller and the right side large. The flow resistance of the hole changes, making the pressure loss on the left side small, the pressure large, the pressure loss on the right side large and the pressure small, so that the oil hydraulic pressure at the two ends of the main valve core is different, driving the main valve core to move to the right, generating load flow and driving the load operation. At the same time, the armature rotates, driving the feedback rod fixed on the armature to shift to the left. Deformation results in counter-clockwise feedback moment. The motion of the valve core makes the feedback rod more deformed and the feedback moment correspondingly larger. When the feedback moments generated by the two are superimposed on the flapper and balanced with the electromagnetic moments generated by the torque motor, the spool is in a predetermined position, and the flapper is in a balanced state. At this time, the required load flow and pressure are generated, and the servo valve is in a predetermined working state. When the load changes or external disturbance causes the spool to deviate from the balanced position, the feedback moment changes, which makes the flapper deviate from the balanced position, and the flow resistance of the variable throttle hole changes accordingly. The pressure difference between the two sides of the spool is generated again, so that the flapper moves in the direction of reducing the deviation until the spool reaches the balanced state [17].

**Figure 1.** Principle diagram of jet flapper valve. (**a**) Structure schematic diagram of servo valve. (**b**) Pre-stage flow field and boundary conditions.

The structural parameters of the nozzle flapper chamber and the set working oil parameters are both as shown in Table 1. According to the working conditions, the maximum working pressure of the servo valve is less than 31 MPa, and the rated flow rate is 0.48–6.9 L/min. The inlet pressure is selected according to the inlet speed. The outlet pressure is set atmospheric pressure.


**Table 1.** Nozzle parameters of the flapper nozzle valve.

The pre-stage valve in this study is perpendicular to the nozzle and is mainly affected by the force in the *x*-axis, i.e., the horizontal direction. While the force in the *y*-axis, i.e., the direction perpendicular to the paper surface, counteracts each other, and has little influence on the servo valve, it can be simulated by two-dimensional flow chart, which saves on computational resources. Moreover, it is convenient to observe the flow field changes in the channel, and it has no effect on the final simulation results. The following grid and simulation are based on the 2D model shown in Figure 1b.
