Analysis of Transient Turbulent Flow in the Pilot Stage of a Deflector Jet Servo Valve

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In this paper, the transient results of the large eddy simulation (LES)-based numerical simulation of the pilot stage flow field are presented. The LES results are validated against particle image velocimetry, showing that the LES method can predict the transient turbulent flow. The length of the potential core of the jet assuming from the nozzle and the position of the jet impinging on the V-groove have great relations with the inlet pressure. Traveling cavitation in the deflector begins to appear when the inlet pressure reaches 6 MPa. The increment in inlet pressure enhances cavitation and cavitation shedding. Snapshot proper orthogonal decomposition(snapshot POD) analysis, based on LES, is applied to decompose the instantaneous velocity fluctuation into coherent structures and turbulent velocity components. The relationships between each coherent structure and wall attachment flow, cavitation phenomena, and vortex pairs are discussed. The vortex in Mode 2 contributes to deflecting the jet from the nozzle and causing part of the fluid to flow along the side wall in the V-groove. Through the fast Fourier transform (FFT) result of the POD coefficient, it is known that the traveling cavitation near the side wall of the V-groove is related to Mode 4. This research contributes to the mathematical modeling and improving the stability of the flow field. cavitation coherent structure computational fluid dynamics deflector jet servo valve V-groove Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 Figure 12 Figure 13 Figure 14 Figure 15 Figure 16 Figure 17 1. Introduction Servo valves are key components in electro-hydraulic servo control systems, are widely used in aerospace, military, agricultural, and other industries [1,2] . It can control the direction and size of the hydraulic oil flow by changing the opening of the valve to achieve accurate control of the hydraulic system. Compared to the jet pipe servo valve, the deflection jet servo valve has the advantage of high response [3] . A typical two-stage servo valve is composed of a torque motor, a deflector jet pilot stage, and a spool valve. The deflector jet pilot stage plays an important role that it converts a small electrical signal into a pressure difference at the ends of the spool [4,5] . Many scholars, discussed below, have paid attention to the pilot stage and conducted a substantial amount of studies. In the deflector jet pilot stage, the flow filed is complicated since there is a submerged jet, an impact jet, energy exchange, and wall-attached flow around the deflector. The flow field is a critical issue and has significant influence on control precision of the spool and working performance of the whole valve. It is, therefore, necessary to undertake further research on the characteristics of turbulent flow. Limited to the test conditions of the micro-flow field, Computational Fluid Dynamics（CFD） has become a common approach because of its low cost and high precision [6] . Y. Zhang [7,8] has conducted CFD simulations to study the pressure and flow characteristics of the pilot stage of servo valve. The influence of structure parameters and boundary conditions on the output characteristics of the pilot stage has been studied by Pham [9] and Somashekhar [10] . In recent years, CFD simulation had been widely used to study cavitation phenomena and erosion wear [11-13] . Hazem K. Abdallah [14] has studied the pressure pulsation characteristics in the flow field of a deflector jet pilot stage by the Large Eddy Simulation(LES) method. Apart from CFD simulation, particle-image-velocimetry (PIV) experiments and mathematical modeling have been conducted to research the pilot-stage flow of the servo valve. Through the two-dimensional-PIV technique, Jianjun Hu [15] has observed the flow field inside a jet pipe servo valve at different deflection angles and flow rates. It was found that increasing the flow rate intensified the vortex scale. Yuesong Li [16] has developed a mathematical model for deflector jet servo valves, which considers structure parameters and reflected physical mechanisms. Kang Shuo [17] has established a flow distribution model based on wall attachment jet theory to analyze the influence of a deflector jet servo valve’s structure on the internal flow distribution. Zhichuang Chen [18] has developed a mathematical model of a deflector jet pilot stage considering boundary layer flow in the V-groove. However, no authors have worked on the detailed influence of the vortex of the unsteady turbulence flow in the deflector. To obtain a reduced-order description of turbulent flow, the POD method combined with LES was utilized in some research. D. Cavar [19] has conducted this method to visualize details of the relationship between the counter-rotating vortex pair, the hanging vortex, and the wake vortices of the jet-in-crossflow. C.I. Chan [20] has used POD to investigate the streamwise turbulent velocity fluctuation of an array of miniature vortex generators. Despite some researchers’ efforts to investigate the flow characteristics in the deflector jet pilot stage, the unsteady performance of vortexes in the deflector has not been studied. Those vortexes increase the instability of the pilot stage flow field and affect the working performance of the valve. In this study, the velocity obtained by the PIV experiment are sufficiently consistent with the LES results. After snapshot POD analysis was conducted to address the velocity field obtained from LES, instantaneous velocity field was decomposed into big-scale coherent structures and small-scale vortexes, based on the triple velocity decomposition. The study discusses how the vortexes influence the flow characteristics and their relationships with different flow phenomena that occurred in the deflector. 2. Working principle of deflector jet servo valve and dimensional geometry of pilot stage The structure of a two-stage deflector jet servo valve is shown in Fig. 1 . When no current passes through the coils, the deflector remains in the middle position. The flow into both receivers are the same, which makes the spool stay in the original position. When there is current through the coils, the torque motor drives the deflector to move. The pressures in the two receivers are different and the spool is promoted to move by the pressure difference. At the same time, the deflector moves in reverse through a feedback rod. Meanwhile, the spool produces a certain flow. The core parts of the deflector jet pilot stage are the deflector and the jet pan. The structural parameters of the pilot stage flow field are listed in Table 1 . Table 1 Structural parameters of the pilot stage flow field. Parameters Symbol Dimension Width of the outlet of the nozzle 2 b 0 0.16mm Distance from outlet of the nozzle to the upper surface of the deflector L c 1 0.11mm Width of the inlet of the V-groove 2 b 2 0.52mm Width of the outlet of the V-groove 2 b 1 0.14mm Thickness of the deflector L w 0.64mm Distance from the outlet of V-groove to the shunt wedge L c 2 0.11mm Width of the shunt wedge l p 0.1mm Width of the receiver inlet l r 0.4mm Half angle between the two receivers α r 1 28◦ Thickness of the jet pan b r 0.4mm In the pilot stage flow field, the high-pressure jet enters the jet pan and issues from the nozzle, where the pressure energy is converted to kinetic energy. As shown in Fig. 2 , the free submerged jet has an increasing cross-section area. The velocity at the outlet of the nozzle remains constant within the potential core. In the mixing zone, the velocity profile is presented as a normal distribution. Then the centerline velocity begins to decreases continually when the distance exceeds the length of the potential core. After the high-speed jet reaches the V-groove, most of the fluid flows downstream to generate a high-pressure area, and a part of it impinges on the wall of the V-groove, where the impact jet, wall attachment flow, and reflux are generated. The flow pattern transforms from free jet into inside-groove flow. During this process, the total average kinetic energy changes after the jet impacts the deflector. The average kinetic energy in the middle is temporarily unchanged, and the velocity direction and magnitude change after the streamlines on both sides touch the wall of the V-groove. The velocity at the vertical boundary is lost, leaving only the velocity along the side boundary. Then fluid flows from the exit of the V-groove and becomes a secondary, free submerged jet. At the same time, the pressure energy of the inside-groove flow is converted into the kinetic energy of the secondary jet. As described above, the flow in the deflector is important to the energy transformation and it is necessary to study it further to improve the performance of the pilot stage. 3. Computational fluid dynamics model 3.1 Geometric model and mesh generation The three-dimensional structure of the pilot stage flow field is rather complicated. To reduce the computational cost while ensuring computational accuracy, the model was simplified so that some nonsignificant oil passages were neglected. As shown in Fig. 3 (a), the detailed three-dimensional components close to the jet area that affect the fluid flow were modeled. We focused on the symmetric flow field when the deflector was in the neutral position, so half of the total model was built to decrease the node number and save computation time, as shown in Fig. 3 (b). Through the meshing module ICEM within ANSYS, all-hexahedral mesh generation for the flow channel was completed, as shown in Fig. 4 . In this work, the origin of coordinates was set in the middle of the shunt wedge; the z -axis was in the depthwise direction, the y-axis was in the streamwise direction, and the x-axis was in the lateral direction. In view of flow characteristics and structures in the deflector jet pilot stage, the V-groove and the shunt wedge had a significant influence on the boundary layer flow. So the mesh near walls of the V-groove and the shunt wedge in the 𝑥 and 𝑦 directions were refined. 3.2 Boundray conditions Four boundaries were defined: the inlet, the outlet, the symmetry, and the wall, as shown in Fig. 3 (b). The inlet of the model was defined as inlet pressure boundary condition, while the outlet as outlet pressure boundary condition. During the simulation, the inlet pressure was set at 4, 6, 10 and 14 MPa, respectively, whereas the outlet pressure was 0.1 MPa. The symmetry plane of the total model was defined as symmetry boundary. Other surfaces in the computational domain were defined as nonslip walls. 3.3 Solution methods and convergence criteria The LES Smagorinsky SGS model approach was adopted to resolve timewise and spacewise variation in the flow [22–23] . During simulations, the SIMPLE scheme, PRESTO!, the bounded central difference, the first-order upwind are applied for pressure-velocity coupling, pressure-discretization, momentum discretization, and the volume fraction. The convergence criteria were set with residuals less than 10 − 5 . The time step was set to 1 × 10 − 5 s according to the feature length and feature velocity in the nozzle. The density and viscosity of liquid phase are 850 Kg/m 3 , 0.00391Kg/(m·s), while those of vapor phase are 0.025 Kg/m 3 , 1×10 − 5 Kg/(m·s). The saturated vapor pressure is set to 3000 Pa. 3.4 Mesh independence In the CFD simulations, the quality of the meshes employed has an effect on the numerical resolution. To verify the mesh independence, the mean values of 2,000 time steps of mass flow rate Q and vapor volume fraction V in the central plane with an inlet pressure of 14 MPa are obtained. The relationship between mesh quantity and above monitoring parameters is shown in Fig. 5 . It can be seen that Q and V both increase with mesh quantity. However, the difference between Meshes 2 and 3 was very small. To save computing cost, we used Mesh 2 in this work. 4. Experimental setup and results comparison with LES The experimental setup comprised PIV, a pump, an adjustable throttle valve, a flow meter, a storage tank, and a deflector pilot stage, as shown in Fig. 6 . As the size of the pilot stage flow field was small, the structure of the test model was scaled up 20 times and simplified to some extent. The tracer particles in the PIV experiment could not distribute evenly in the hydraulic oil, so water was used as the medium in the test. To make the model similar to the real flow conditions, the Reynolds similarity criterion was used to determine the size of the model and the flow rate of water. Reynolds similarity criterion is expressed as $$\frac{{ \rho }_{p}{v}_{p}{l}_{p}}{{\mu }_{p}}=\frac{{\rho }_{m}{v}_{m}{l}_{m}}{{\mu }_{m}}=Re$$ 1 Here, ρ , v , l , µ represent flow density, velocity, feature length, and dynamic viscosity, respectively; p represents the prototype of the valve; and m is the test model. The main equipment used for PIV experiments included a dynamic studio, a CCD camera, a synchronizer controller, and a sheet laser. To meet the requirements of the PIV experiment, the material of the test model was high-translucency plexiglass, which had a refraction index close to water. To visualize the fluid motion, the water was seeded with fluorescent particles (diameter 10 µm). An Nd:YLF laser was chosen to illuminate the center of the test model in a depthwise direction. In the pilot stage, the jet strength and the turbulence intensity of the central plane (z = 0) were the greatest. In this work, we focused on analyzing the transient flow in the central plane. To verify the accuracy of the simulation data, PIV experiments were conducted in the same working conditions with LES at an inlet pressure of 10 MPa. The velocity distribution data at positions y / n = 3.75 were extracted, as shown in Fig. 7 . The results of LES and experiments were approximately close to each other. Differences appeared to be caused by the identification of boundary layer motion. On accounting for the measuring error of PIV, LES results were utilized to research the pilot-stage flow field in this study. 5. Results and discussions of simulations 5.1 Analysis of free submerged jet and collision with the side wall Between the fluid assuming from the nozzle and the fluid in the surrounding static environment, there was a velocity discontinuity, which fluctuated due to instability and developed into vortexes, so that the fluid in the surrounding static environment was sucked into the jet. The turbulent jet structure was divided into two parts: the initial section and the main section. The initial section consisted of two regions: the potential core, where the fluid velocity was not affected by the surrounding static fluid, and the mixing zone, where the velocity distribution was similar to that of the main body segment. Figure 8 shows the velocity distribution along the jet center line and the side wall of the V-groove. From Fig. 8 (a), the length of the potential core L 0 can be obtained, which presented the rightmost end of the curve to the marked point. On the wall of the V-groove, a velocity peak existed, which was the position of the center jet. The length of the potential core and the position of the center jet impinging on the V-groove were the same when the inlet pressure was lower than 6 MPa and then increased a little when the inlet pressure increased. 5.2 Effect of inlet pressure on cavitation phenomenon For the deflector jet pilot stage, a large number of vortexes exist because of the complicated structure of the flow channel. When the pressure at the vortex is lower than the saturated vapor pressure, cavitation occurs. The influence factor of cavitation phenomena and its influence mechanisms should be studied, as cavitation reduces the working performance of the pilot stage. As a key influence factor, the inlet pressure was set to 4, 6, 10, and 14 MPa, respectively. When inlet pressure was 4 MPa, the cavitation phenomena were not evident. The transient distributions of the vapor phase volume fraction for different inlet pressures in the research zone were shown in Figs. 9 –11. Figure 11 Transient cavitation phenomena with time step 2×10 − 5 s at inlet pressure 14Mpa It can be seen that cavitation phenomena exist at the inlet of the V-groove. The intensity of cavitation became greater when the inlet pressure increased. A small part of the fluid flowed out alone along the side wall and then flowed over a sharp corner to generate traveling cavitation. The process of formation, growth, falling off, and collapse of traveling cavitation was evident when the inlet pressure was above 10 MPa. So the flow field in the pilot stage oscillated, partly due to the cavitation phenomena in the V-groove. 6. LES based snapshot POD analysis As wall attachment flow, boundary layer flow, and cavitation phenomena had significant relationships with vortexes in the deflector, how these vortexes influence the performance of the flow field were studied in the following sections. The snapshot POD analysis were mentioned in the previous researches [24-27] . To better verify the snapshot number independence, the energy ratio of the first four modes in the V-groove is listed in Table 2. It is known that the energy ratios of Mode 1 and Mode 3 increase with increasing snapshot numbers, whereas those of Mode 2 and Mode 4 decrease. The differences between 400 and 500 snapshots are negligible. In this work, transient datasets of 400 snapshots were chosen as the basis for snapshot POD analysis. Table2. Results of the snapshot number independence study Snapshot number 300 400 500 600 Energy ratio of mode 1 0.92290 0.92315 0.92344 0.92451 Energy ratio of mode 2 0.02339 0.02330 0.02329 0.02327 Energy ratio of mode 3 0.00761 0.00767 0.00768 0.00770 Energy ratio of mode 4 0.00550 0.00540 0.00538 0.00537 6.2 Analysis of snapshot POD of velocity field For ease of analysis, the origin of the coordinate system of the flow field was fixed at the center of the shunt wedge, and the streamwise (y) and lateral (x) distances from the origin were normalized by the nozzle length ( n ). The flow filed of the deflector is located in the region of 0 < y / n < 4.7, where the velocity data from LES are conducted by snapshot POD. The snapshot POD results of the first five modes were obtained, as shown in Figure 12. Mode 1 consists of the mean flow field with most energy. From snapshot POD results of different inlet pressures, it is shown that the velocity distribution of the first modes is very close, whereas differences in the higher modes are clearly observed. As inlet pressure increases, the energy of Mode 1 decreases, and the energy of the high modes increases. The high modes contain coherent flow fields and turbulent flow fields, which are related to the instability of the flow fields. To determine the truncation order between the coherent flow field and the turbulent flow field, this paper defined the correlation coefficient between the reconstructed flow field obtained by selecting the number of adjacent modes as: Figure 13 shows the relationship between the correlation coefficient of the reconstructed flow field and the number of corresponding adjacent modes. With the increase in mode number, the correlation coefficient starts to increase rapidly. In either case, the correlation coefficient between the flow field reconstructed from Mode 2 to Mode 5 and the flow field reconstructed from Mode 2 to Mode 6 was above 95%. The rate of increase in the correlation coefficient became very slow when the mode number continued to increase. The result shows that starting from Mode 6, the vortex structure of the higher mode had a very weak effect on the first five modes, so the boundary truncation order of the coherent structure in the pilot stage could be determined as five. The flow field reconstructed from Modes 2–5 could characterize the coherent flow field of the transient flow field. Since the coherent flow field carried a large proportion of the turbulent kinetic energy, most information about the turbulent pulsation flow field could be described from the flow field reconstructed from Modes 2–5. To further study the influence of coherent structures, streamlines from Modes 2–5 for inlet pressures of 10 MPa and 14 MPa are presented in Figure 14. In Mode 2, a large-scale wall-attached vortex existed near the downstream part of the side wall of the V-groove, while the vortex core was located at 2 < y / n < 2.4. With the inlet pressure increasing, the locations of vortex cores increased in the y direction. At the inlet of the V-groove, the free submerged jet was deflected to the wall side and then attached to the wall while flowing downward. It could be seen that this jet deflection was due to the effect of the big-scale clockwise vortex in Mode 2. In Mode 3, a counter-rotating vortex pair (a small-scale wall-attached vortex and a counterclockwise vortex) became the major factor. Snapshot POD analysis results showed a full ability to directly visualize details of the counter-rotating vortex pair. In Mode 4, in the upper right region near the wall, there was a vortex, which was close to the position of the traveling cavitation in Part 5.2. Whether they are rated is discussed further. With the mode increasing, the vortexes became smaller and more turbulent, and their positions rose with the inlet pressure increasing. 6. 4 Analysis of POD coefficients of coherent structures Since the POD mode is related to the corresponding flow field structure, we can deeply study the influence of coherent structures of various scales through the spectral analysis of mode coefficients. Figure 15 showed the FFT results of the corresponding coefficients of the first four modes at different inlet pressures, which meaned the pulsation frequency of each mode. When the inlet pressure was 6 MPa, the frequency domain of a 1 , a 2 , a3, and a 4 peaked at 3,380 Hz, 1,170 Hz, 976 Hz, and 3,710 Hz, respectively. For inlet pressure 10 MPa, the frequency domain of a 1 , a 2 , a3, and a 4 peaked at 2,343 Hz, 976 Hz, 976 Hz, and 1,367 Hz, respectively, while at 2,832 Hz, 1,269 Hz, 1,269 Hz, and 2,832 Hz for 14 MPa. Figure 16 showed the FFT results of surface mean velocity in the V-groove. Compared to Figure 15, the frequencies were very close to those of the first modes, which meaned that the oscillation in the V-groove primarily came from the first mode with the most energy. The traveling cavitation caused the pressure to fluctuate. Point A (0.2 mm, 0.7 mm, 0) was located near the vortex core of the traveling cavitation. The pressure at Point A varied with time, as shown in Figure 17(a). The pressure fluctuated more when the inlet pressure increased. To verify the assumption of the relationship between the vortex in Mode 4 and the traveling cavitation, the FFT analysis of pressure at Point A was conducted, as shown in Figures 17(b) and (c). The FFT results of 1,376 Hz and 2,834 Hz were close to those of a 4 for 10 and 14 MPa, which were 1,367 Hz and 2,832 Hz, respectively. It could be concluded that the traveling cavitation was generated by the low-pressure vortex in Mode 4. 7. Conclusion Investigations aimed at analyzing the turbulent flow and understanding the complex flow patterns in the deflector jet pilot stage were conducted by combining the LES method with the snapshot POD technique. The validity of the LES method was verified by PIV experiments. When the inlet pressure was lower than 6 MPa, the length of the potential core of the jet assuming from the nozzle, and the position of the center jet impinging on the V-groove had little relationship with the inlet pressure. Cavitation phenomena were also not evident. All elements were affected when the inlet pressure increased. The process of formation, growth, falling off, and collapse of traveling cavitation occurred when the inlet pressure was above 10 MPa. By using the snapshot POD technique, eigenfunctions of the velocity field calculated by LES method under different inlet pressures were obtained in the current study. The energy of Mode 1 decreased as inlet pressure increased, but the energy of the higher modes increased. The velocity eigenfunctions of the Mode 1 were similar, while the locations and strengths of the vortexes in higher modes were different for different inlet pressures. The correlation coefficient between the reconstructed flow fields of adjacent modes was obtained to extract coherent structures. The results of the LES method and snapshot POD showed that the positions of the vortex in Mode 4 and cavitation were close. Through FFT analysis, the frequency domain results of the snapshot POD coefficients of coherent structures were obtained. By comparing the FFT results, it could be concluded that the traveling cavitation had a relationship with Mode 4. The oscillation in the V-groove primarily came from Mode 1 with the most energy. In Mode 2, a big-scale clockwise vortex affected the jet near the inlet of the V-groove, which had an influence on energy transformation. In Mode 3, a counter-rotating vortex pair (a small-scale wall-attached vortex and a counterclockwise vortex) also rose with the increasing inlet pressure. Mode 5, with a small vortex, became more turbulent compared to the lower modes. The above coherent structures were related to the parameter of inlet pressure and affected the flow characteristics of the flow field in the deflector jet pilot stage. The transient turbulent flow in the deflector is complicate that it is difficult to study its flow characteristics. Combining LES with snapshot POD is an effective way to study how each coherent structure contributes to the different flow phenomena that occurred in the deflector. 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Zhang W , Ge Y J , Cai C S .Application of Snapshot POD Analysis in Extracting Flow Structures around Bridge Decks[J].Advances in Structural Engineering, 2015, 18(6):803-815. Cite Share Download PDF Status: Published Journal Publication published 11 Apr, 2025 Read the published version in Journal of the Brazilian Society of Mechanical Sciences and Engineering → Version 1 posted Reviewers agreed at journal 20 May, 2024 Reviewers invited by journal 05 May, 2024 Editor assigned by journal 21 Apr, 2024 First submitted to journal 18 Apr, 2024 Editorial decision: Major revisions 31 Mar, 2024 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. 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Wu","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA0ElEQVRIiWNgGAWjYFAC5oYDDBUScvwSYJ6EDBFaGIFazlgYS84AsoBaeIjSwsDYUpG44QZYCwNhLQY3EhsPFzZIGBvfbj7+6EaNBQ8D++GjGwhoaTg8c4eEnNmdY4nNOceADuNJS7uBT4sZSAvvGQljsxs5hs05bEAtEjxmRGhpk0jcPAOk5R8pWjZIALXkthGhxf7Mw4bDPECHSdxIS5yd2yfBw0bIL5LtyYc/81TUyfHPSD7wOecbkMF++BheLZiAjTTlo2AUjIJRMAqwAQDMGEwsprSkPQAAAABJRU5ErkJggg==","orcid":"","institution":"Wuhan University of Science and Technology","correspondingAuthor":true,"prefix":"","firstName":"Lin","middleName":"","lastName":"Wu","suffix":""},{"id":298898992,"identity":"edb17354-97bc-4819-b906-7e57c72bed82","order_by":1,"name":"Xian Zou","email":"","orcid":"","institution":"Wuhan University of Science and Technology","correspondingAuthor":false,"prefix":"","firstName":"Xian","middleName":"","lastName":"Zou","suffix":""}],"badges":[],"createdAt":"2024-03-26 08:19:55","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4168187/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4168187/v1","draftVersion":[],"editorialEvents":[{"content":"https://doi.org/10.1007/s40430-025-05542-9","type":"published","date":"2025-04-11T16:04:57+00:00"}],"editorialNote":"","failedWorkflow":false,"files":[{"id":56410632,"identity":"bc71c417-8ff8-4fd1-9122-e0aaf31e4e87","added_by":"auto","created_at":"2024-05-13 20:21:23","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":105016,"visible":true,"origin":"","legend":"\u003cp\u003eStructure of a deflector jet servo valve\u003csup\u003e[20]\u003c/sup\u003e\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-4168187/v1/25ad4fafa6641b8b58e22a80.png"},{"id":56410630,"identity":"0b949375-34d8-4be1-9437-566c63ffe61e","added_by":"auto","created_at":"2024-05-13 20:21:23","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":10843,"visible":true,"origin":"","legend":"\u003cp\u003eFree submerged jet in pilot stage\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-4168187/v1/fda6e725c4c1a03c0c84083d.png"},{"id":56410631,"identity":"1db453e7-9b14-413c-9573-541cd3893c79","added_by":"auto","created_at":"2024-05-13 20:21:23","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":27074,"visible":true,"origin":"","legend":"\u003cp\u003eTotal geometric model and boundary conditions\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-4168187/v1/d27a11a46c7e3abdb818fb98.png"},{"id":56411697,"identity":"e7796da9-1139-423d-b65f-613966dec841","added_by":"auto","created_at":"2024-05-13 20:29:24","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":43973,"visible":true,"origin":"","legend":"\u003cp\u003eMesh generation\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-4168187/v1/3297f7eddc47631f8dca983d.png"},{"id":56410635,"identity":"5929ab87-e792-4c6e-b5d3-31e294883816","added_by":"auto","created_at":"2024-05-13 20:21:23","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":41554,"visible":true,"origin":"","legend":"\u003cp\u003eThe mesh independence.\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-4168187/v1/0964767b4aee7d6fcb7a322d.png"},{"id":56410634,"identity":"96dbe2f9-b5f0-4d5f-a5d7-5f4843417098","added_by":"auto","created_at":"2024-05-13 20:21:23","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":142556,"visible":true,"origin":"","legend":"\u003cp\u003ePIV testing rig with CCD camera and Nd:YLF laser\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-4168187/v1/290b8fbaf666b93b0b6d5053.png"},{"id":56410644,"identity":"f75fe429-d219-4cdc-bab3-5054002f6497","added_by":"auto","created_at":"2024-05-13 20:21:24","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":11178,"visible":true,"origin":"","legend":"\u003cp\u003eReliability verification: Comparison of velocity distribution at the position \u003cem\u003ey\u003c/em\u003e/\u003cem\u003en\u003c/em\u003e=3.75 PIV and LES\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-4168187/v1/00957dd784f1b1ad15047571.png"},{"id":56410642,"identity":"e211097b-c4de-4fe9-920a-ba1b32661c37","added_by":"auto","created_at":"2024-05-13 20:21:24","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":28283,"visible":true,"origin":"","legend":"\u003cp\u003e(a) Velocity distribution along the jet center line, (b) Velocity distribution along the side wall of the V-groove\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-4168187/v1/987e1fd32efcdcf06ad60b90.png"},{"id":56410639,"identity":"30eb38d0-ec20-4d68-8fd8-9d0683b87977","added_by":"auto","created_at":"2024-05-13 20:21:24","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":18432,"visible":true,"origin":"","legend":"\u003cp\u003eTransient cavitation phenomena with time step 2×10\u003csup\u003e-5\u003c/sup\u003es at inlet pressure 6Mpa\u003c/p\u003e","description":"","filename":"9.png","url":"https://assets-eu.researchsquare.com/files/rs-4168187/v1/e194983c111977a36d3c3e4e.png"},{"id":56410641,"identity":"b6eededd-0cc1-4d97-8ff7-661379b207ce","added_by":"auto","created_at":"2024-05-13 20:21:24","extension":"png","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":26086,"visible":true,"origin":"","legend":"\u003cp\u003eTransient cavitation phenomena with time step 2×10\u003csup\u003e-5\u003c/sup\u003es at inlet pressure 10Mpa\u003c/p\u003e","description":"","filename":"10.png","url":"https://assets-eu.researchsquare.com/files/rs-4168187/v1/099558b3700fe1d6f667d657.png"},{"id":56410645,"identity":"a2245a9b-9075-48cf-bf96-59551e610578","added_by":"auto","created_at":"2024-05-13 20:21:24","extension":"png","order_by":11,"title":"Figure 11","display":"","copyAsset":false,"role":"figure","size":26827,"visible":true,"origin":"","legend":"\u003cp\u003eTransient cavitation phenomena with time step 2×10\u003csup\u003e-5\u003c/sup\u003es at inlet pressure 14Mpa\u003c/p\u003e","description":"","filename":"11.png","url":"https://assets-eu.researchsquare.com/files/rs-4168187/v1/644ca83cfa94eea875182567.png"},{"id":56410646,"identity":"b5bfa931-8fc8-4df7-b975-41d8fad42e5c","added_by":"auto","created_at":"2024-05-13 20:21:24","extension":"png","order_by":12,"title":"Figure 12","display":"","copyAsset":false,"role":"figure","size":265581,"visible":true,"origin":"","legend":"\u003cp\u003eThe first five modes for different inlet pressures, (a) 6Mpa, (b)10Mpa, (c) 14Mpa\u003c/p\u003e","description":"","filename":"12.png","url":"https://assets-eu.researchsquare.com/files/rs-4168187/v1/a5ec52da71a48b4c095e11ad.png"},{"id":56410637,"identity":"73064b2e-6d3b-41de-963b-ea37264d9c88","added_by":"auto","created_at":"2024-05-13 20:21:23","extension":"png","order_by":13,"title":"Figure 13","display":"","copyAsset":false,"role":"figure","size":14452,"visible":true,"origin":"","legend":"\u003cp\u003eCorrelation coefficient of the reconstructed flow field by adjacent modes\u003c/p\u003e","description":"","filename":"13.png","url":"https://assets-eu.researchsquare.com/files/rs-4168187/v1/717412e34723e46a2c0472a7.png"},{"id":56410647,"identity":"bb71fa7c-2e6e-4abd-8ae3-bdd815f457eb","added_by":"auto","created_at":"2024-05-13 20:21:24","extension":"png","order_by":14,"title":"Figure 14","display":"","copyAsset":false,"role":"figure","size":180567,"visible":true,"origin":"","legend":"\u003cp\u003eStreamlines in the V-groove( inlet pressure are 10, 14Mpa from left to right)\u003c/p\u003e","description":"","filename":"14.png","url":"https://assets-eu.researchsquare.com/files/rs-4168187/v1/08d16590b0d9f13d05a610b2.png"},{"id":56410638,"identity":"c5f042db-0e55-4585-88b6-61d3a65c0fdf","added_by":"auto","created_at":"2024-05-13 20:21:24","extension":"png","order_by":15,"title":"Figure 15","display":"","copyAsset":false,"role":"figure","size":126843,"visible":true,"origin":"","legend":"\u003cp\u003eFrequency domain analysis of POD coefficients of the first four modes.\u003c/p\u003e","description":"","filename":"15.png","url":"https://assets-eu.researchsquare.com/files/rs-4168187/v1/89365f3e6aa45989e040a5dd.png"},{"id":56410640,"identity":"d5e5c4e7-7c12-41ed-812a-6d6ad71b4247","added_by":"auto","created_at":"2024-05-13 20:21:24","extension":"png","order_by":16,"title":"Figure 16","display":"","copyAsset":false,"role":"figure","size":58571,"visible":true,"origin":"","legend":"\u003cp\u003eFFT results of surface mean velocity in the V-groove (inlet pressure are 6, 10, 14Mpa from left to right)\u003c/p\u003e","description":"","filename":"16.png","url":"https://assets-eu.researchsquare.com/files/rs-4168187/v1/8587e16943ccdaa4ace75c43.png"},{"id":56410643,"identity":"9b897c8f-aeda-44ad-adda-cad70eee97b4","added_by":"auto","created_at":"2024-05-13 20:21:24","extension":"png","order_by":17,"title":"Figure 17","display":"","copyAsset":false,"role":"figure","size":54442,"visible":true,"origin":"","legend":"\u003cp\u003e(a) Pressure at Point A varies with time, (b) FFT analysis of pressure at Point A with inlet pressure 10Mpa, (c) FFT analysis of pressure at Point A with inlet pressure 14Mpa.\u003c/p\u003e","description":"","filename":"17.png","url":"https://assets-eu.researchsquare.com/files/rs-4168187/v1/ad13375bd1dfd5208bf53b26.png"},{"id":80559102,"identity":"05f18ade-0494-4f76-9041-b270261908d9","added_by":"auto","created_at":"2025-04-14 16:17:48","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":1941676,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4168187/v1/a25026b8-ab98-42a3-955d-b822b753c18c.pdf"}],"financialInterests":"","formattedTitle":"Analysis of Transient Turbulent Flow in the Pilot Stage of a Deflector Jet Servo Valve","fulltext":[{"header":"1.\tIntroduction","content":"\u003cp\u003eServo valves are key components in electro-hydraulic servo control systems, are widely used in aerospace, military, agricultural, and other industries\u003csup\u003e[1,2]\u003c/sup\u003e. It can control the direction and size of the hydraulic oil flow by changing the opening of the valve to achieve accurate control of the hydraulic system. Compared to the jet pipe servo valve, the deflection jet servo valve has the advantage of high response\u003csup\u003e[3]\u003c/sup\u003e. A typical two-stage servo valve is composed of a torque motor, a deflector jet pilot stage, and a spool valve. The deflector jet pilot stage plays an important role that it converts a small electrical signal into a pressure difference at the ends of the spool\u003csup\u003e[4,5]\u003c/sup\u003e. Many scholars, discussed below, have paid attention to the pilot stage and conducted a substantial amount of studies. In the deflector jet pilot stage, the flow filed is complicated since there is a submerged jet, an impact jet, energy exchange, and wall-attached flow around the deflector. The flow field is a critical issue and has significant influence on control precision of the spool and working performance of the whole valve. It is, therefore, necessary to undertake further research on the characteristics of turbulent flow.\u003c/p\u003e\n\u003cp\u003eLimited to the test conditions of the micro-flow field, Computational Fluid Dynamics（CFD）\u0026nbsp;has become a common approach because of its low cost and high precision\u003csup\u003e[6]\u003c/sup\u003e.\u0026nbsp;Y. Zhang\u003csup\u003e[7,8]\u0026nbsp;\u003c/sup\u003ehas conducted CFD simulations to study the pressure and flow characteristics of the pilot stage of servo valve. The influence of structure parameters and boundary conditions on the output characteristics of the pilot stage has been studied by Pham\u003csup\u003e[9]\u003c/sup\u003e and Somashekhar\u003csup\u003e[10]\u003c/sup\u003e. In recent years, CFD simulation had been widely used to study cavitation phenomena and erosion wear\u003csup\u003e[11-13]\u003c/sup\u003e. Hazem K. Abdallah\u003csup\u003e[14]\u003c/sup\u003e has studied the pressure pulsation characteristics in the flow field of a deflector jet pilot stage by the\u0026nbsp;Large Eddy Simulation(LES) method. Apart from CFD simulation,\u0026nbsp;particle-image-velocimetry (PIV) experiments\u0026nbsp;and\u0026nbsp;mathematical modeling have been conducted to research the pilot-stage flow of the servo valve. Through the two-dimensional-PIV technique, Jianjun Hu\u003csup\u003e[15]\u003c/sup\u003e has observed the flow field inside a jet pipe servo valve at different deflection angles and flow rates.\u0026nbsp;It was found that increasing the flow rate intensified the vortex scale. Yuesong Li\u003csup\u003e[16]\u003c/sup\u003e has developed a mathematical model for deflector jet servo valves, which considers structure parameters and reflected physical mechanisms. Kang Shuo\u003csup\u003e[17]\u003c/sup\u003e has established a flow distribution model based on wall attachment jet theory to analyze the influence of a deflector jet servo valve\u0026rsquo;s structure on the internal flow distribution.\u0026nbsp;Zhichuang Chen\u003csup\u003e[18]\u003c/sup\u003e has developed a mathematical model of a deflector jet pilot stage considering boundary layer flow in the V-groove. However, no authors have worked on the detailed influence of the vortex of the unsteady turbulence flow in the deflector.\u003c/p\u003e\n\u003cp\u003eTo obtain a reduced-order description of turbulent flow, the POD method combined with LES was utilized in some research. D. Cavar\u003csup\u003e[19]\u003c/sup\u003e has conducted this method to visualize details of the relationship between the counter-rotating vortex pair, the hanging vortex, and the wake vortices of the jet-in-crossflow. C.I. Chan\u003csup\u003e[20]\u003c/sup\u003e has used POD to investigate the streamwise turbulent velocity fluctuation of an array of miniature vortex generators.\u003c/p\u003e\n\u003cp\u003eDespite some researchers\u0026rsquo; efforts to investigate the flow characteristics in the deflector jet pilot stage, the unsteady performance of vortexes in the deflector has not been studied. Those vortexes increase the instability of the pilot stage flow field and affect the working performance of the valve. In this study, the velocity obtained by the PIV experiment are sufficiently consistent with the LES results. After snapshot POD analysis was conducted to address the velocity field obtained from LES, instantaneous velocity field was decomposed into big-scale coherent structures and small-scale vortexes, based on the triple velocity decomposition. The study discusses how the vortexes influence the flow characteristics and their relationships with different flow phenomena that occurred in the deflector.\u0026nbsp;\u003c/p\u003e"},{"header":"2. Working principle of deflector jet servo valve and dimensional geometry of pilot stage","content":"\u003cp\u003eThe structure of a two-stage deflector jet servo valve is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig1\" class=\"InternalRef\"\u003e1\u003c/span\u003e. When no current passes through the coils, the deflector remains in the middle position. The flow into both receivers are the same, which makes the spool stay in the original position. When there is current through the coils, the torque motor drives the deflector to move. The pressures in the two receivers are different and the spool is promoted to move by the pressure difference. At the same time, the deflector moves in reverse through a feedback rod. Meanwhile, the spool produces a certain flow. The core parts of the deflector jet pilot stage are the deflector and the jet pan. The structural parameters of the pilot stage flow field are listed in Table\u0026nbsp;\u003cspan refid=\"Tab1\" class=\"InternalRef\"\u003e1\u003c/span\u003e.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003e \u003cdiv class=\"gridtable\"\u003e\u003ctable float=\"Yes\" id=\"Tab1\" border=\"1\"\u003e \u003ccaption language=\"En\"\u003e \u003cdiv class=\"CaptionNumber\"\u003eTable 1\u003c/div\u003e \u003cdiv class=\"CaptionContent\"\u003e \u003cp\u003eStructural parameters of the pilot stage flow field.\u003c/p\u003e \u003c/div\u003e \u003c/caption\u003e \u003ccolgroup cols=\"3\"\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c1\" colnum=\"1\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c2\" colnum=\"2\"\u003e\u003c/div\u003e \u003cdiv align=\"left\" class=\"colspec\" colname=\"c3\" colnum=\"3\"\u003e\u003c/div\u003e \u003cthead\u003e \u003ctr\u003e \u003cth align=\"left\" colname=\"c1\"\u003e \u003cp\u003eParameters\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c2\"\u003e \u003cp\u003eSymbol\u003c/p\u003e \u003c/th\u003e \u003cth align=\"left\" colname=\"c3\"\u003e \u003cp\u003eDimension\u003c/p\u003e \u003c/th\u003e \u003c/tr\u003e \u003c/thead\u003e \u003ctbody\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eWidth of the outlet of the nozzle\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e2\u003cem\u003eb\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.16mm\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDistance from outlet of the nozzle to the upper surface of the deflector\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003eL\u003c/em\u003e\u003csub\u003e\u003cem\u003ec\u003c/em\u003e1\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.11mm\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eWidth of the inlet of the V-groove\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e2\u003cem\u003eb\u003c/em\u003e\u003csub\u003e2\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.52mm\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eWidth of the outlet of the V-groove\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e2\u003cem\u003eb\u003c/em\u003e\u003csub\u003e1\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.14mm\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eThickness of the deflector\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003eL\u003c/em\u003e\u003csub\u003e\u003cem\u003ew\u003c/em\u003e\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.64mm\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eDistance from the outlet of V-groove to the shunt wedge\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003eL\u003c/em\u003e\u003csub\u003e\u003cem\u003ec\u003c/em\u003e2\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.11mm\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eWidth of the shunt wedge\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003el\u003c/em\u003e\u003csub\u003e\u003cem\u003ep\u003c/em\u003e\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.1mm\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eWidth of the receiver inlet\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003el\u003c/em\u003e\u003csub\u003e\u003cem\u003er\u003c/em\u003e\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.4mm\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eHalf angle between the two receivers\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003eα\u003c/em\u003e\u003csub\u003e\u003cem\u003er\u003c/em\u003e1\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e28◦\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003ctr\u003e \u003ctd align=\"left\" colname=\"c1\"\u003e \u003cp\u003eThickness of the jet pan\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c2\"\u003e \u003cp\u003e\u003cem\u003eb\u003c/em\u003e\u003csub\u003e\u003cem\u003er\u003c/em\u003e\u003c/sub\u003e\u003c/p\u003e \u003c/td\u003e \u003ctd align=\"left\" colname=\"c3\"\u003e \u003cp\u003e0.4mm\u003c/p\u003e \u003c/td\u003e \u003c/tr\u003e \u003c/tbody\u003e \u003c/colgroup\u003e \u003c/table\u003e\u003c/div\u003e \u003c/p\u003e \u003cp\u003eIn the pilot stage flow field, the high-pressure jet enters the jet pan and issues from the nozzle, where the pressure energy is converted to kinetic energy. As shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig2\" class=\"InternalRef\"\u003e2\u003c/span\u003e, the free submerged jet has an increasing cross-section area. The velocity at the outlet of the nozzle remains constant within the potential core. In the mixing zone, the velocity profile is presented as a normal distribution. Then the centerline velocity begins to decreases continually when the distance exceeds the length of the potential core. After the high-speed jet reaches the V-groove, most of the fluid flows downstream to generate a high-pressure area, and a part of it impinges on the wall of the V-groove, where the impact jet, wall attachment flow, and reflux are generated. The flow pattern transforms from free jet into inside-groove flow. During this process, the total average kinetic energy changes after the jet impacts the deflector. The average kinetic energy in the middle is temporarily unchanged, and the velocity direction and magnitude change after the streamlines on both sides touch the wall of the V-groove. The velocity at the vertical boundary is lost, leaving only the velocity along the side boundary. Then fluid flows from the exit of the V-groove and becomes a secondary, free submerged jet. At the same time, the pressure energy of the inside-groove flow is converted into the kinetic energy of the secondary jet. As described above, the flow in the deflector is important to the energy transformation and it is necessary to study it further to improve the performance of the pilot stage.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e"},{"header":"3. Computational fluid dynamics model","content":"\u003cdiv id=\"Sec4\" class=\"Section2\"\u003e \u003ch2\u003e3.1 Geometric model and mesh generation\u003c/h2\u003e \u003cp\u003eThe three-dimensional structure of the pilot stage flow field is rather complicated. To reduce the computational cost while ensuring computational accuracy, the model was simplified so that some nonsignificant oil passages were neglected. As shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e(a), the detailed three-dimensional components close to the jet area that affect the fluid flow were modeled. We focused on the symmetric flow field when the deflector was in the neutral position, so half of the total model was built to decrease the node number and save computation time, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e(b). Through the meshing module ICEM within ANSYS, all-hexahedral mesh generation for the flow channel was completed, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig4\" class=\"InternalRef\"\u003e4\u003c/span\u003e. In this work, the origin of coordinates was set in the middle of the shunt wedge; the \u003cem\u003ez\u003c/em\u003e-axis was in the depthwise direction, the y-axis was in the streamwise direction, and the x-axis was in the lateral direction.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eIn view of flow characteristics and structures in the deflector jet pilot stage, the V-groove and the shunt wedge had a significant influence on the boundary layer flow. So the mesh near walls of the V-groove and the shunt wedge in the \u0026#119909; and \u0026#119910; directions were refined.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec5\" class=\"Section2\"\u003e \u003ch2\u003e3.2 Boundray conditions\u003c/h2\u003e \u003cp\u003eFour boundaries were defined: the inlet, the outlet, the symmetry, and the wall, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig3\" class=\"InternalRef\"\u003e3\u003c/span\u003e(b). The inlet of the model was defined as inlet pressure boundary condition, while the outlet as outlet pressure boundary condition. During the simulation, the inlet pressure was set at 4, 6, 10 and 14 MPa, respectively, whereas the outlet pressure was 0.1 MPa. The symmetry plane of the total model was defined as symmetry boundary. Other surfaces in the computational domain were defined as nonslip walls.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec6\" class=\"Section2\"\u003e \u003ch2\u003e3.3 Solution methods and convergence criteria\u003c/h2\u003e \u003cp\u003eThe LES Smagorinsky SGS model approach was adopted to resolve timewise and spacewise variation in the flow\u003csup\u003e[22\u0026ndash;23]\u003c/sup\u003e. During simulations, the SIMPLE scheme, PRESTO!, the bounded central difference, the first-order upwind are applied for pressure-velocity coupling, pressure-discretization, momentum discretization, and the volume fraction. The convergence criteria were set with residuals less than 10\u003csup\u003e\u0026minus;\u0026thinsp;5\u003c/sup\u003e. The time step was set to 1 \u0026times; 10\u003csup\u003e\u0026minus;\u0026thinsp;5\u003c/sup\u003es according to the feature length and feature velocity in the nozzle. The density and viscosity of liquid phase are 850 Kg/m\u003csup\u003e3\u003c/sup\u003e, 0.00391Kg/(m\u0026middot;s), while those of vapor phase are 0.025 Kg/m\u003csup\u003e3\u003c/sup\u003e, 1\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;5\u003c/sup\u003e Kg/(m\u0026middot;s). The saturated vapor pressure is set to 3000 Pa.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec7\" class=\"Section2\"\u003e \u003ch2\u003e3.4 Mesh independence\u003c/h2\u003e \u003cp\u003eIn the CFD simulations, the quality of the meshes employed has an effect on the numerical resolution. To verify the mesh independence, the mean values of 2,000 time steps of mass flow rate \u003cem\u003eQ\u003c/em\u003e and vapor volume fraction \u003cem\u003eV\u003c/em\u003e in the central plane with an inlet pressure of 14 MPa are obtained. The relationship between mesh quantity and above monitoring parameters is shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig5\" class=\"InternalRef\"\u003e5\u003c/span\u003e. It can be seen that \u003cem\u003eQ\u003c/em\u003e and \u003cem\u003eV\u003c/em\u003e both increase with mesh quantity. However, the difference between Meshes 2 and 3 was very small. To save computing cost, we used Mesh 2 in this work.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003c/div\u003e"},{"header":"4. Experimental setup and results comparison with LES","content":"\u003cp\u003eThe experimental setup comprised PIV, a pump, an adjustable throttle valve, a flow meter, a storage tank, and a deflector pilot stage, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig6\" class=\"InternalRef\"\u003e6\u003c/span\u003e. As the size of the pilot stage flow field was small, the structure of the test model was scaled up 20 times and simplified to some extent. The tracer particles in the PIV experiment could not distribute evenly in the hydraulic oil, so water was used as the medium in the test. To make the model similar to the real flow conditions, the Reynolds similarity criterion was used to determine the size of the model and the flow rate of water. Reynolds similarity criterion is expressed as\u003cdiv id=\"Equ1\" class=\"Equation\"\u003e\u003cdiv format=\"TEX\" class=\"mathdisplay\" id=\"FileID_Equ1\" name=\"EquationSource\"\u003e\n$$\\frac{{ \\rho }_{p}{v}_{p}{l}_{p}}{{\\mu }_{p}}=\\frac{{\\rho }_{m}{v}_{m}{l}_{m}}{{\\mu }_{m}}=Re$$\u003c/div\u003e\u003cdiv class=\"EquationNumber\"\u003e1\u003c/div\u003e\u003c/div\u003e\u003c/p\u003e \u003cp\u003eHere, \u003cem\u003eρ\u003c/em\u003e, \u003cem\u003ev\u003c/em\u003e, \u003cem\u003el\u003c/em\u003e, \u003cem\u003e\u0026micro;\u003c/em\u003e represent flow density, velocity, feature length, and dynamic viscosity, respectively; \u003cem\u003ep\u003c/em\u003e represents the prototype of the valve; and \u003cem\u003em is\u003c/em\u003e the test model.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e \u003cp\u003eThe main equipment used for PIV experiments included a dynamic studio, a CCD camera, a synchronizer controller, and a sheet laser. To meet the requirements of the PIV experiment, the material of the test model was high-translucency plexiglass, which had a refraction index close to water. To visualize the fluid motion, the water was seeded with fluorescent particles (diameter 10 \u0026micro;m). An Nd:YLF laser was chosen to illuminate the center of the test model in a depthwise direction. In the pilot stage, the jet strength and the turbulence intensity of the central plane (z\u0026thinsp;=\u0026thinsp;0) were the greatest. In this work, we focused on analyzing the transient flow in the central plane.\u003c/p\u003e \u003cp\u003eTo verify the accuracy of the simulation data, PIV experiments were conducted in the same working conditions with LES at an inlet pressure of 10 MPa. The velocity distribution data at positions \u003cem\u003ey\u003c/em\u003e/\u003cem\u003en\u003c/em\u003e\u0026thinsp;=\u0026thinsp;3.75 were extracted, as shown in Fig.\u0026nbsp;\u003cspan refid=\"Fig7\" class=\"InternalRef\"\u003e7\u003c/span\u003e. The results of LES and experiments were approximately close to each other. Differences appeared to be caused by the identification of boundary layer motion. On accounting for the measuring error of PIV, LES results were utilized to research the pilot-stage flow field in this study.\u003c/p\u003e \u003cp\u003e \u003c/p\u003e"},{"header":"5. Results and discussions of simulations","content":"\u003cdiv id=\"Sec10\" class=\"Section2\"\u003e\n \u003ch2\u003e5.1 Analysis of free submerged jet and collision with the side wall\u003c/h2\u003e\n \u003cp\u003eBetween the fluid assuming from the nozzle and the fluid in the surrounding static environment, there was a velocity discontinuity, which fluctuated due to instability and developed into vortexes, so that the fluid in the surrounding static environment was sucked into the jet. The turbulent jet structure was divided into two parts: the initial section and the main section. The initial section consisted of two regions: the potential core, where the fluid velocity was not affected by the surrounding static fluid, and the mixing zone, where the velocity distribution was similar to that of the main body segment. Figure \u003cspan class=\"InternalRef\"\u003e8\u003c/span\u003e shows the velocity distribution along the jet center line and the side wall of the V-groove. From Fig. \u003cspan class=\"InternalRef\"\u003e8\u003c/span\u003e(a), the length of the potential core \u003cem\u003eL\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e can be obtained, which presented the rightmost end of the curve to the marked point. On the wall of the V-groove, a velocity peak existed, which was the position of the center jet. The length of the potential core and the position of the center jet impinging on the V-groove were the same when the inlet pressure was lower than 6 MPa and then increased a little when the inlet pressure increased.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec11\" class=\"Section2\"\u003e\n \u003ch2\u003e5.2 Effect of inlet pressure on cavitation phenomenon\u003c/h2\u003e\n \u003cp\u003eFor the deflector jet pilot stage, a large number of vortexes exist because of the complicated structure of the flow channel. When the pressure at the vortex is lower than the saturated vapor pressure, cavitation occurs. The influence factor of cavitation phenomena and its influence mechanisms should be studied, as cavitation reduces the working performance of the pilot stage.\u003c/p\u003e\n \u003cp\u003eAs a key influence factor, the inlet pressure was set to 4, 6, 10, and 14 MPa, respectively. When inlet pressure was 4 MPa, the cavitation phenomena were not evident. The transient distributions of the vapor phase volume fraction for different inlet pressures in the research zone were shown in Figs. \u003cspan class=\"InternalRef\"\u003e9\u003c/span\u003e\u0026ndash;11.\u003c/p\u003e\n \u003cp\u003eFigure\u0026nbsp;11 Transient cavitation phenomena with time step 2\u0026times;10\u003csup\u003e\u0026minus;\u0026thinsp;5\u003c/sup\u003es at inlet pressure 14Mpa\u003c/p\u003e\n \u003cp\u003eIt can be seen that cavitation phenomena exist at the inlet of the V-groove. The intensity of cavitation became greater when the inlet pressure increased. A small part of the fluid flowed out alone along the side wall and then flowed over a sharp corner to generate traveling cavitation. The process of formation, growth, falling off, and collapse of traveling cavitation was evident when the inlet pressure was above 10 MPa. So the flow field in the pilot stage oscillated, partly due to the cavitation phenomena in the V-groove.\u003c/p\u003e\n\u003c/div\u003e"},{"header":"6. LES based snapshot POD analysis","content":"\u003cp\u003eAs wall attachment flow, boundary layer flow, and cavitation phenomena had significant relationships with vortexes in the deflector, how these vortexes influence the performance of the flow field were studied in the following sections.\u003c/p\u003e\n\u003cp\u003eThe snapshot POD analysis were mentioned in the previous researches\u003csup\u003e[24-27]\u003c/sup\u003e. To better verify the snapshot number independence, the energy ratio of the first four modes in the V-groove is listed in Table 2. It is known that the energy ratios of Mode 1 and Mode 3 increase with increasing snapshot numbers, whereas those of Mode 2 and Mode 4 decrease. The differences between 400 and 500 snapshots are negligible. In this work, transient datasets of 400 snapshots were chosen as the basis for snapshot POD analysis.\u003c/p\u003e\n\u003cp\u003eTable2. Results of the snapshot number independence study\u0026nbsp;\u003c/p\u003e\n\u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd width=\"34.623655913978496%\" valign=\"top\"\u003e\n \u003cp\u003eSnapshot number\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.344086021505376%\" valign=\"top\"\u003e\n \u003cp\u003e300\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.344086021505376%\" valign=\"top\"\u003e\n \u003cp\u003e400\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.344086021505376%\" valign=\"top\"\u003e\n \u003cp\u003e500\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.344086021505376%\" valign=\"top\"\u003e\n \u003cp\u003e600\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"34.623655913978496%\" valign=\"top\"\u003e\n \u003cp\u003eEnergy ratio of mode 1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.344086021505376%\" valign=\"top\"\u003e\n \u003cp\u003e0.92290\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.344086021505376%\" valign=\"top\"\u003e\n \u003cp\u003e0.92315\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.344086021505376%\" valign=\"top\"\u003e\n \u003cp\u003e0.92344\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.344086021505376%\" valign=\"top\"\u003e\n \u003cp\u003e0.92451\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"34.623655913978496%\" valign=\"top\"\u003e\n \u003cp\u003eEnergy ratio of mode 2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.344086021505376%\" valign=\"top\"\u003e\n \u003cp\u003e0.02339\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.344086021505376%\" valign=\"top\"\u003e\n \u003cp\u003e0.02330\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.344086021505376%\" valign=\"top\"\u003e\n \u003cp\u003e0.02329\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.344086021505376%\" valign=\"top\"\u003e\n \u003cp\u003e0.02327\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"34.623655913978496%\" valign=\"top\"\u003e\n \u003cp\u003eEnergy ratio of mode 3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.344086021505376%\" valign=\"top\"\u003e\n \u003cp\u003e0.00761\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.344086021505376%\" valign=\"top\"\u003e\n \u003cp\u003e0.00767\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.344086021505376%\" valign=\"top\"\u003e\n \u003cp\u003e0.00768\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.344086021505376%\" valign=\"top\"\u003e\n \u003cp\u003e0.00770\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd width=\"34.623655913978496%\" valign=\"top\"\u003e\n \u003cp\u003eEnergy ratio of mode 4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.344086021505376%\" valign=\"top\"\u003e\n \u003cp\u003e0.00550\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.344086021505376%\" valign=\"top\"\u003e\n \u003cp\u003e0.00540\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.344086021505376%\" valign=\"top\"\u003e\n \u003cp\u003e0.00538\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd width=\"16.344086021505376%\" valign=\"top\"\u003e\n \u003cp\u003e0.00537\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e6.2\u0026nbsp;Analysis of snapshot POD of velocity field\u003c/p\u003e\n\u003cp\u003eFor ease of analysis, the origin of the coordinate system of the flow field was fixed at the center of the shunt wedge, and the streamwise (y) and lateral (x) distances from the origin were normalized by the nozzle length (\u003cem\u003en\u003c/em\u003e). The flow filed of the deflector is located in the region of 0 \u003cem\u003e\u0026lt; y\u003c/em\u003e/\u003cem\u003en \u0026lt;\u0026nbsp;\u003c/em\u003e4.7, where the velocity data from LES are conducted by snapshot POD. The snapshot POD results of the first five modes were obtained, as shown in Figure 12.\u003c/p\u003e\n\u003cp\u003eMode 1 consists of the mean flow field with most energy. From snapshot POD results of different inlet pressures, it is shown that the velocity distribution of the first modes is very close, whereas differences in the higher modes are clearly observed. As inlet pressure increases, the energy of Mode 1 decreases, and the energy of the high modes increases. The high modes contain coherent flow fields and turbulent flow fields, which are related to the instability of the flow fields.\u003c/p\u003e\n\u003cp\u003eTo determine the truncation order between the coherent flow field and the turbulent flow field, this paper defined the correlation coefficient between the reconstructed flow field obtained by selecting the number of adjacent modes as:\u003c/p\u003e\n\u003cp\u003e\u003cimg 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\"\u003e\u003cbr\u003e\u003c/p\u003e\n\u003cp\u003eFigure 13 shows the relationship between the correlation coefficient of the reconstructed flow field and the number of corresponding adjacent modes. With the increase in mode number, the correlation coefficient starts to increase rapidly. In either case, the correlation coefficient between the flow field reconstructed from Mode 2 to Mode 5 and the flow field reconstructed from Mode 2 to Mode 6 was above 95%. The rate of increase in the correlation coefficient became very slow when the mode number continued to increase. The result shows that starting from Mode 6, the vortex structure of the higher mode had a very weak effect on the first five modes, so the boundary truncation order of the coherent structure in the pilot stage could be determined as five. The flow field reconstructed from Modes 2\u0026ndash;5 could characterize the coherent flow field of the transient flow field. Since the coherent flow field carried a large proportion of the turbulent kinetic energy, most information about the turbulent pulsation flow field could be described from the flow field reconstructed from Modes 2\u0026ndash;5.\u003c/p\u003e\n\u003cp\u003eTo further study the influence of coherent structures, streamlines from Modes 2\u0026ndash;5 for inlet pressures of 10 MPa and 14 MPa are presented in Figure 14.\u003c/p\u003e\n\u003cp\u003eIn Mode 2, a large-scale wall-attached vortex existed near the downstream part of the side wall of the V-groove, while the vortex core was located at 2 \u0026lt;\u003cem\u003e\u0026nbsp;y\u003c/em\u003e/\u003cem\u003en\u003c/em\u003e \u0026lt; 2.4. With the inlet pressure increasing, the locations of vortex cores increased in the \u003cem\u003ey\u003c/em\u003e direction. At the inlet of the V-groove, the free submerged jet was deflected to the wall side and then attached to the wall while flowing downward. It could be seen that this jet deflection was due to the effect of the big-scale clockwise vortex in Mode 2. In Mode 3, a counter-rotating vortex pair (a small-scale wall-attached vortex and a counterclockwise vortex) became the major factor. Snapshot POD analysis results showed a full ability to directly visualize details of the counter-rotating vortex pair. In Mode 4, in the upper right region near the wall, there was a vortex, which was close to the position of the traveling cavitation in Part 5.2. Whether they are rated is discussed further. With the mode increasing, the vortexes became smaller and more turbulent, and their positions rose with the inlet pressure increasing.\u003c/p\u003e\n\u003cp\u003e6. 4 Analysis of\u0026nbsp;POD coefficients\u0026nbsp;of coherent structures\u003c/p\u003e\n\u003cp\u003eSince the POD mode is related to the corresponding flow field structure, we can deeply study the influence of coherent structures of various scales through the spectral analysis of mode coefficients. Figure 15 showed the FFT results of the corresponding coefficients of the first four modes at different inlet pressures, which meaned the pulsation frequency of each mode. When the inlet pressure was 6 MPa, the frequency domain of a\u003csub\u003e1\u003c/sub\u003e, a\u003csub\u003e2\u003c/sub\u003e, a3, and a\u003csub\u003e4\u003c/sub\u003e peaked at 3,380 Hz, 1,170 Hz, 976 Hz, and 3,710 Hz, respectively. For inlet pressure 10 MPa, the frequency domain of a\u003csub\u003e1\u003c/sub\u003e, a\u003csub\u003e2\u003c/sub\u003e, a3, and a\u003csub\u003e4\u003c/sub\u003e peaked at 2,343 Hz, 976 Hz, 976 Hz, and 1,367 Hz, respectively, while at 2,832 Hz, 1,269 Hz, 1,269 Hz, and 2,832 Hz for 14 MPa.\u003c/p\u003e\n\u003cp\u003eFigure 16 showed the FFT results of surface mean velocity in the V-groove. Compared to Figure 15, the frequencies were very close to those of the first modes, which meaned that the oscillation in the V-groove primarily came from the first mode with the most energy.\u003c/p\u003e\n\u003cp\u003e\u0026nbsp;The traveling cavitation caused the pressure to fluctuate. Point A (0.2 mm, 0.7 mm, 0) was located near the vortex core of the traveling cavitation. The pressure at Point A varied with time, as shown in Figure 17(a). The pressure fluctuated more when the inlet pressure increased. To verify the assumption of the relationship between the vortex in Mode 4 and the traveling cavitation, the FFT analysis of pressure at Point A was conducted, as shown in Figures 17(b) and (c). The FFT results of 1,376 Hz and 2,834 Hz were close to those of a\u003csub\u003e4\u003c/sub\u003e for 10 and 14 MPa, which were 1,367 Hz and 2,832 Hz, respectively. It could be concluded that the traveling cavitation was generated by the low-pressure vortex in Mode 4. \u0026nbsp;\u003c/p\u003e"},{"header":" 7. Conclusion","content":"\u003cp\u003eInvestigations aimed at analyzing the turbulent flow and understanding the complex flow patterns in the deflector jet pilot stage were conducted by combining the LES method with the snapshot POD technique. The validity of the LES method was verified by PIV experiments. When the inlet pressure was lower than 6 MPa, the length of the potential core of the jet assuming from the nozzle, and the position of the center jet impinging on the V-groove had little relationship with the inlet pressure. Cavitation phenomena\u0026nbsp;were also not evident. All elements were affected when\u0026nbsp;the inlet pressure increased. The process of formation, growth, falling off, and collapse of traveling cavitation occurred when the inlet pressure was above 10 MPa.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eBy using the snapshot POD technique, eigenfunctions of the velocity field calculated by LES method under different inlet pressures were obtained in the current study. The energy of Mode 1 decreased as inlet pressure increased, but the energy of the higher modes increased. The velocity eigenfunctions of the Mode 1 were similar, while the locations and strengths of the vortexes in higher modes were different for different inlet pressures. The correlation coefficient between the reconstructed flow fields of adjacent modes was obtained to extract coherent structures.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eThe results of the LES method and snapshot POD showed that the positions of the vortex in Mode 4 and cavitation were close. Through FFT analysis, the frequency domain results of the snapshot POD coefficients of coherent structures were obtained. By comparing the FFT results, it could be concluded that the traveling cavitation had a relationship with Mode 4. The oscillation in the V-groove\u0026nbsp;\u003ca href=\"javascript%3A;\"\u003eprimarily\u003c/a\u003e came\u0026nbsp;\u003ca href=\"javascript%3A;\"\u003efrom\u003c/a\u003e Mode 1 with the most energy. In Mode 2, a big-scale clockwise vortex affected the jet near the inlet of the V-groove, which had an influence on energy transformation. In Mode 3, a counter-rotating vortex pair (a small-scale wall-attached vortex and a counterclockwise vortex) also rose with the increasing inlet pressure. Mode 5, with a small vortex, became more turbulent compared to the lower modes. The above coherent structures were related to the parameter of inlet pressure and affected the flow characteristics of the flow field in the deflector jet pilot stage.\u003c/p\u003e\n\u003cp\u003eThe transient turbulent flow in the deflector is complicate that it is difficult to study its flow characteristics. Combining LES with snapshot POD is an effective way to study how each coherent structure contributes to the different flow phenomena that occurred in the deflector. It is helpful to prove position control of the deflector and establish mathematical model of the complicate flow filed.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003eFunding：This work was funded by the Hubei Provincial Natural Science Foundation of China, grant number 2021CFB141, and numerical calculation is supported by the High-Performance Computing Center of Wuhan University of Science and Technology.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n\u003cli\u003eYu Wang, Yaobao Yin. Performance reliability of jet pipe servo valve under random vibration environment[J]. Mechatronics, 2019, 64(2):102286.\u003c/li\u003e\n\u003cli\u003eLingkang Meng, Yuchuan Zhu, Defa Wu, Jianjun Ding. Experimental and simulation research of performance degradation of jet pipe servo valve under wedge erosion of jet amplifier[J]. Engineering Failure Analysis, 2023,154: 107602.\u003c/li\u003e\n\u003cli\u003eLin Wu, Kuisheng Chen, Yuan Guo. 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In this paper, the transient results of the large eddy simulation (LES)-based numerical simulation of the pilot stage flow field are presented. The LES results are validated against particle image velocimetry, showing that the LES method can predict the transient turbulent flow. The length of the potential core of the jet assuming from the nozzle and the position of the jet impinging on the V-groove have great relations with the inlet pressure. Traveling cavitation in the deflector begins to appear when the inlet pressure reaches 6 MPa. The increment in inlet pressure enhances cavitation and cavitation shedding. Snapshot proper orthogonal decomposition(snapshot POD) analysis, based on LES, is applied to decompose the instantaneous velocity fluctuation into coherent structures and turbulent velocity components. The relationships between each coherent structure and wall attachment flow, cavitation phenomena, and vortex pairs are discussed. The vortex in Mode 2 contributes to deflecting the jet from the nozzle and causing part of the fluid to flow along the side wall in the V-groove. Through the fast Fourier transform (FFT) result of the POD coefficient, it is known that the traveling cavitation near the side wall of the V-groove is related to Mode 4. This research contributes to the mathematical modeling and improving the stability of the flow field.\u003c/p\u003e","manuscriptTitle":"Analysis of Transient Turbulent Flow in the Pilot Stage of a Deflector Jet Servo Valve","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-05-13 20:21:18","doi":"10.21203/rs.3.rs-4168187/v1","editorialEvents":[{"type":"communityComments","content":0},{"type":"reviewerAgreed","content":"","date":"2024-05-20T05:39:10+00:00","index":0,"fulltext":""},{"type":"reviewersInvited","content":"","date":"2024-05-05T14:14:33+00:00","index":"","fulltext":""},{"type":"editorAssigned","content":"","date":"2024-04-21T17:45:11+00:00","index":"","fulltext":""},{"type":"submitted","content":"Journal of the Brazilian Society of Mechanical Sciences and Engineering","date":"2024-04-18T19:57:09+00:00","index":"","fulltext":""},{"type":"decision","content":"Major revisions","date":"2024-03-31T20:06:23+00:00","index":"","fulltext":""}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"journal-of-the-brazilian-society-of-mechanical-sciences-and-engineering","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"bmse","sideBox":"Learn more about [Journal of the Brazilian Society of Mechanical Sciences and Engineering](http://link.springer.com/journal/40430)","snPcode":"40430","submissionUrl":"https://www.editorialmanager.com/bmse/default2.aspx","title":"Journal of the Brazilian Society of Mechanical Sciences and Engineering","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false}}],"origin":"","ownerIdentity":"eb44bc64-1103-432e-911a-66816a945f8e","owner":[],"postedDate":"May 13th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"published-in-journal","subjectAreas":[],"tags":[],"updatedAt":"2025-04-14T16:15:14+00:00","versionOfRecord":{"articleIdentity":"rs-4168187","link":"https://doi.org/10.1007/s40430-025-05542-9","journal":{"identity":"journal-of-the-brazilian-society-of-mechanical-sciences-and-engineering","isVorOnly":false,"title":"Journal of the Brazilian Society of Mechanical Sciences and Engineering"},"publishedOn":"2025-04-11 16:04:57","publishedOnDateReadable":"April 11th, 2025"},"versionCreatedAt":"2024-05-13 20:21:18","video":"","vorDoi":"10.1007/s40430-025-05542-9","vorDoiUrl":"https://doi.org/10.1007/s40430-025-05542-9","workflowStages":[]},"version":"v1","identity":"rs-4168187","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4168187","identity":"rs-4168187","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}