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Radial Flow Force at the Annular Orifice of a Two-Dimensional Hydraulic Servo Valve

A two-dimensional hydraulic servo valve is an innovative servo control element that provides hydraulic systems with a high power-to-weight ratio, great anti-pollution potential, and superior static and dynamic characteristics. The spool of such a valve is subject to two degrees of freedom: rotation around and sliding along the spool axis, to accomplish both pilot control and flow amplifier functions. The structure of the spool at the main stage is similar to that of a traditional slide valve wherein the asymmetrical distribution of the oil paths manufactured in the valve body produces circumferential unevenness in the radial flow force to the spool at the annular orifice, in line with the momentum theory. Three-dimensional computational fluid dynamics analysis of the flow field revealed that the radial flow force at the annular orifice increases sharply with inlet flow velocity and decreases as the orifice opening grows, while changes in outlet pressure do not affect the levels or distribution of this force. Also, net radial force at the annular orifice increases with both inlet velocity and opening size. The paper presents results demonstrating that the net radial force from fluid flow through the orifice could increase friction resistance and cannot be safely ignored, especially under high-flow-rate conditions.


I. INTRODUCTION
A two-dimensional (2D) hydraulic servo valve is a control unit whose fast dynamic response and capacity for precise flow control play a critical role in determining the overall performance of the hydraulic servo system. This type of servo valve represents a novel solution for pilot control of a two-stage valve, combining the pressure control handled by the pilot stage and the flow amplification through the main stage into a single-spool operation. Its inventor, Ruan [1], proved that a 2D servo valve overcomes the drawbacks related to a nozzle flapper servo valve's sensitivity to oil contamination and a jet pipe servo valve's high pilot leakage. He established the stability criteria and analysed the structural parameters that affect the static and dynamic characteristics of a 2D servo flow valve [2]. Meanwhile, the nonlinearities, such as electromagnetic saturation and hysteresis, are greatly reduced The associate editor coordinating the review of this manuscript and approving it for publication was Tao Wang . by application of a stage control method to the electrical-tomechanical actuator [3]. Li et al. [4] implemented a tracking control algorithm for the electro-mechanical converter, which helps enhance the dynamic characteristics of the 2D servo valve: the −3 dB frequency of the valve was 210 Hz, and the rise time for step response was approximately 8 ms.
A 2D servo valve with both rotary and linear motions on a single spool offers the benefits of a high powerto-weight ratio, high anti-pollution capacity, and response sensitivity. Alongside this type of valve, such components as 2D high-speed on-off valves [5] and 2D proportional hydraulic valves [6], [7] employ the same operation mechanism, designed to expand the branching of 2D hydraulic valves and enrich hydraulic control units' products. For further improving the power-to-weight ratio and improving the efficiency of 2D servo valves, it is essential to study the flow forces and frictional resistance acting on the spool, which are balanced with the driving force generated by the electro-hydraulic actuator.
The steady flow force is the force acting on the spool because of fluid flow through the valve, which has an important role in the control of high-hydraulic-power systems [8], [9]. Yuan and Li [8] studied fluid force in efforts to increase the agility of the spool such that fast spool response can be achieved without as much power from the actuator. They found that steady flow force can be reduced via the viscosity and non-metering flux effect. In other work, Herakovič [9] implemented geometric modification of the sliding spool and the valve socket to compensate for the axial flow forces by designing a conical form for the sliding spool at the inlet control edge, which helped to decrease fluid velocity at the outlet cross-section. Then, studying the flow force acting on the spool of a directional-control valve, Rajda and Lisowski [10] verified that the flow has very little influence on the axial flow force for the inflow path while being highly significant for the outflow path. In more recent research, Frosina et al. [11], [12] focused on simulation and calculation of axial flow forces within directional-control valves by comparing the stream lines under various spool displacement conditions and inlet flow rates. Their results indicated that the force values contingent on spool displacement create strong force oscillations when the spool was moved from zero to 2.8 mm.
Amirante et al. [13] confirmed the necessity of full three-dimensional (3D) computational fluid dynamics (CFD) investigation of fluid flow through proportional-type hydraulic directional-control valves, which can accurately predict the flow forces acting on the spool in aims of minimising the required driving forces. They also redesigned the central conical and lateral surfaces of the new spool and succeeded in reducing the actuation force by more than 10% from that of the commercial spool commonly used in four-way three-position directly operated valves at maximal opening for the given flow rate examined in the experimental verification [14]. Han et al. [15] studied the axial flow forces on the spool with a double U-groove structure of a cartridge valve. They found that the highest axial flow forces decreased by 10% with their groove-geometry optimisation. Xie et al. [16], in turn, investigated the steady axial flow force compensation of a converged-flow cartridge proportional valve. They optimised the spool structure by adding a damping tail to regulate the flow force acting on the spool and to diminish the negative impact of flow force without affecting the pressure drop between the inlet and outlet. In recent work, Li et al. [17] designed a damping flange for the spool of the same cartridge valve to compensate for the axial flow force. The optimal parameters for the damping flange were obtained via CFD analysis, and experimental testing revealed a 93% decrease in flow force after the damping flange determined to be optimal was implemented. Related work was carried out by Qu et al. [18], who reduced the steady flow force through their design of a turbine bucket profile in the spool groove of the electro-hydraulic proportional relief valve. The flow force increment with their optimised spool was smaller than that experienced with the original spool when the flow was increased under various pressures. For instance, the flow force fell by 26.7% when the working pressure was 4 MPa.
Lugowski [19] modified the formulae for the flow forces on the basis of measurements of the pressure distribution near the orifice of spool-type hydraulic valves with and without a compensation structure. Furthermore, the sticking problem experienced by these valves was studied in light of the flow force and its distribution [20], and the motion function of the spool was established as creating non-sticking threshold conditions. Flow force was found to be influenced more by the valve opening than by the temperature, especially for small openings [21].
As the literature review above indicates, previous studies were focused mainly on the axial-flow force component. Many means of reducing or compensating for the axial flow force have been investigated in pursuit of more efficient hydraulic valves and lower energy costs, while the radial flow force has often been neglected or assumed to be axially symmetrical [19]- [21]. The fluid cavity and paths are not, in fact, axially symmetrical within the valve, so the radial flow force at the orifice must be influenced, and these forces do not cancel each other out.
Zhang et al. [22] studied the radial forces acting on the spool of a 2D valve and found this force to affect the spool of the 2D servo valve in three distinct ways -with regard to the cone profile of the spool shoulder, the eccentricity between the spool and sleeve, and the asymmetry of the spiral channels of the pilot stage. Clamping force caused by eccentricity at the pilot stage of the 2D electro-hydraulic proportional directional-control valve was investigated in studies by Liu and colleagues [6], who proved that the radial force can be reduced by cutting a flat plane over each high-pressure and low-pressure hole on the spool shoulder at the pilot stage -but at the expense of increasing leakage. In more recent work, Lu et al. [23] studied the radial flow force induced when the liquid passes through the helical orifices of the pilot stage of a 2D servo valve. They found that this force was affected by the flow rate and the helical orifice opening; moreover, the force increased with the flow rate and acted in the direction opposite the driving force.
The spool of a 2D servo valve is a sliding-type element, and its main stage is in charge of the flow rate, as is the case in most sliding-spool valves. The flow rate is controlled by the annular orifice formed by the spool and housing, and the flow force rises in direct proportion to the flow rate [9], opposite the driving force as noted above. All the above-mentioned studies focused on the characteristics of the axial flow force and on compensation methods to reduce that force so as to improve the ratio of the power to the sliding-valve density further. Researchers have assumed the radial flow forces to cancel themselves out because of the symmetrical structure. Again, the inner flow cavity of the valve is not entirely symmetrical, especially with the plate-mounting type. This is because the four or five oil ports and passages created in the housing are asymmetrically distributed in accordance VOLUME 8, 2020 with the ISO 103772:1992 standard, which was reviewed and confirmed in 2017 [24].
When the fluid flows into the asymmetric fluid cavity and enters the annular orifice, the fluid's pressure and velocity are not uniformly distributed over the orifice, so the radial component of the flow force is not zero. This was verified by Rajda and Lisowski [10], who found the force in the two directions perpendicular to the spool axis to be 40 N and 28 N, for the inlet flow path, and the values were 252 N and 83 N for the outlet flow path, when the flow rate was 450 l/min. Moreover, the momentum theory assumes jet angle θ to be 69 • [25], so the radial flow force expected is 2.6 times the axial flow force [20]. Such studies indicate that the radial force is affected by jet flow through the orifice. Therefore, it is important to examine the radial component of flow force because it has negative effects on the spool, as the frictional force comprises a Coulomb friction component produced by radial forces. The Coulomb friction forces must be avoided, because they could cause the dead-band span of a valve, and consequently demand more force potential from the electro-mechanical actuator. Studying the effect of both axial and radial flow forces on the spool is an important aspect of endeavours to minimise an actuator's power demands.
What this paper contributes to scholarship and practice is numerical examination of the radial flow force and discussion of that force's influence on a 2D servo valve. The results of our study highlight the importance of the radial flow force in the design of both 2D servo valves and traditional sliding valves. If both the axial and radial flow forces were minimised, the actuator's power requirements could be reduced, which would result in a higher power-to-weight ratio.
In the discussion below, the radial flow force at the annular orifice of the main stage in a 2D servo valve is considered, thus: Firstly, the structure and principle of operation are introduced. Then, Section 3 provides an illustration of our numerical model, based on the partial 3D flow domain. The influences of the orifice opening, outlet pressure, and flow rate on the radial flow force are discussed in the fourth section, followed by analysis of the net radial force acting on the spool. The paper concludes with a summary and suggestions for future work.

II. THE TWO-DIMENSIONAL VALVE A. THE STRUCTURE AND PRINCIPLE
The main-stage structure of the spool inside the 2D servo valve studied is similar to that of traditional slide spools, and the five oil passages machined in to the housing are arranged asymmetrically. Therefore, the flow cavity formed by the spool and housing should affect the radial component of flow force at the annular orifice when fluid flows through it. Fig. 1 shows the structure of a 2D servo valve with a nominal size of 22 mm. The pilot stage of the valve is presented at the left in the smaller rectangular box, and the main stage is outlined in the right-hand rectangular area. When the fluid flows into port P, it then fills the annular groove formed by the spool and the housing, and it flows into the hole (c) drilled through the centre rod of the spool. After this, the fluid enters the high-pressure cavity (HPC) and the right cavity (RC), respectively, through a long blind hole within the spool. The opening area of orifice H formed by the HPC and the spiral channel (SC) in null state is identical to that of orifice L, formed by the spiral channel and low-pressure cavity (LPC), so the SC pressure is half the value at port P. As the left cavity (LC) is connected directly to the SC, the pressure of the LC is half the RC's. The circular portion of the left end is designed to be twice the size of the annular area in the RC of the spool, so the axial forces on the spool that are produced by the LC and RC pressure are equal at the null position and the spool is stationary.
When the spool is manipulated anticlockwise into an angle by its actuator, the opening area of orifice H is larger than that of orifice L; thereby, the LC pressure rises. The force at the left end of the spool becomes greater than that at the right acting on the annular surface in the RC. In consequence, the spool slides to the right. Then, the annular orifice formed between ports P and B opens, and another orifice, formed between ports A and T, opens at the same time. This state is maintained until the opening areas of orifices H and L in the pilot stage become the same again. If the spool rotates clockwise, the spool's operation is vice versa.

B. FLOW FORCES ON THE SPOOL AT THE ANNULAR ORIFICE
The annular orifice between ports P and B opens after the spool slides to the right. The fluid control volume of this path has been selected for calculation of the flow force acting on the spool from a jet discharging into port B. Fig. 2 shows a partial schematic diagram of the main stage and the control volume of the spool cavity, which is filled in grey color. This control volume allows fluid to enter the cut groove connected with port P, then pass into the groove connected with port B after the vena contracta. The formulae for the radial and axial steady flow forces caused by the fluid flows across the orifice can be expressed in terms of the following equations (1) and (2) on the basis of the momentum theory [9]: 207940 VOLUME 8, 2020 where F r is the radial flow force, in newtons (N); F a is the axial flow force, also in newtons; C q is the flow coefficient; C v is the velocity coefficient; D is the shoulder diameter of the spool, in metres (m); x is the opening of the annular orifice, in metres; p is the pressure difference, in pascals (Pa); and θ is the jet angle, referring to the acute angle between the velocity at the vena contracta and the X direction.

III. NUMERICAL MODELLING A. GEOMETRY
The fluid domain of the P-B flow path is chosen for analysis of the radial flow force at the annular orifice of the 2D servo valve. The 3D control volume is shown in Fig. 3. Both the inlet and the outlet passage have a diameter of 20 mm, and the diameter of the shoulder and rod of the spool are, respectively, 24 mm and 11 mm. The depth of the groove in the housing is 7 mm, and the width is 16 mm. For the control surface of the spool shoulder that forms the annular orifice with the groove, a conical shape is maintained, per the design shown in Fig. 2. According to Lugowski [19], this conical control surface for the spool shoulder compensates for the axial steady flow force better than a traditional annular plane does. For the model, the unstructured tetrahedral mesh of the computational domain was created via the grid generator of ANSYS Workbench 19.0. To investigate the mesh-independence, we generated four unstructured meshes, with 0.38, 0.87, 2.07, and 6.30 million computation cells, labelled as grids 1 to 4 in Fig. 4, and performed simulation for a valve opening x of 0.5 mm. We compared the target value -the outlet flow rate -across the four grids. The discretisation error was estimated by means of the procedure presented by Celik et al. [26]. The estimated discretisation error bars are shown in Fig. 4, with the error value presented in brackets. This comparison led us to select Grid 3 for the simulation: it yielded similar values for flow rate to the more computationally demanding Grid 4, in addition to which the numerical uncertainty corresponding to the flow rate solution for the fine grid (Grid 3) was only 0.15%. Thus, the findings from grid-independence analysis  (see Fig. 4) led us to choose the mesh strategy of Grid 3 for other valve opening conditions.

B. GOVERNING EQUATIONS AND THE SOLUTION STRATEGIES
The steady-state Reynolds-averaged Navier-Stokes (RANS) equations were solved via the commercial CFD application ANSYS Fluent. The k − ε turbulence model was used to predict turbulence because it accounts for low Reynolds number effects and swirling. The flow was treated as turbulent since the lowest Reynolds number in the domain is 439, which is greater than the critical Reynolds numbers for the spool valve, ranging from 200 to 300 [27]. The enhanced wall treatment (EWT) technique was employed for the work presented here because the y + range for the boundary layers of the relevant mesh is 13 to 27, and EWT is insensitive to the non-dimensional distance y + and covers intermediated y + value in the buffer layer (5 < y + < 30). The governing equations were solved with a pressure-based solver. The SIMPLE and PRESTO! schemes were applied, for the pressure-velocity coupling and pressure discretisation, respectively. We utilised the second-order upwind scheme for handling the momentum, turbulent kinetic energy, and turbulent dissipation equations. The velocity inlet was set to be the inlet boundary condition, and the pressure outlet was specified as the outlet boundary condition. For the density and kinetic viscosity of the fluid (ISO VG 46), we used 889 kg/m 3 and 4.5 × 10 −5 m 2 /s, respectively.
Simulations were conducted for several openings, inlet velocities, and outlet pressure; Table 1 lists the values for the boundary conditions and opening sizes. For the internal flow in the modelling, we regarded the inlet flow as fully developed, so a turbulence intensity and hydraulic diameter specification method was chosen for the calculations. The hydraulic diameter was specified as equal to the diameter of the inlet port, and the value chosen for the turbulence intensity was 6.5 per cent, as the value estimated via the empirical correlation formula, below, varied from 5.2 to 6.5 per cent as the inlet velocity was decreased.
Here, I denotes the turbulence intensity, which is defined as the ratio of the root mean square of the velocity fluctuations, u , to the mean flow velocity, u avg , and R eD H is the Reynolds number.

IV. RESULTS AND DISCUSSION
The radial flow force caused by the fluid flowing through the annular orifice is distributed over the round edge of the spool. Therefore, we selected four critical sections for purposes of collecting the relevant simulation data for pressure and jet angle θ at each position angle α. These are referred to as the 0 • -180 • , 45 • -225 • , 90 • -270 • , and 135 • -315 • sections. They are depicted in Fig. 5.
As the fluid converges into the orifice, pressure drops rapidly as the flow velocity rises. The changes in the velocity's magnitude and direction generate flow force acting on the spool, with prior research [19] pointing to an effective range of 1 mm downstream from the orifice. Therefore, in our work, the pressure value was retrieved for planes P up and P down , maintaining a distance of 0.8 mm upstream and downstream of the control edge of the orifice, respectively, as shown in Fig. 6. Fig. 7 presents the annular distributions of pressures over planes P up and P down and jet angle θ for the two outlet pressures 10 MPa and 5 MPa. The orifice opening is 0.5 mm, and the inlet velocity is 5 m/s.  In Fig. 7, the smallest values for the jet angle are oriented toward the outlet passage direction, where α is 0 • . The jet angle remains constant for the rest of the annular orifice. We calculated the radial flow forces at the various positions in accordance with formula (1), for which the flow coefficient (C q ) and velocity coefficient (C v ) were set to 0.61 and 0.98 [25], respectively. The two curves in Fig. 8 represent the circumferential distribution of the radial flow force under different outlet pressure conditions. These two curves for around the orifice are almost entirely congruent, it is evident from this result that the radial flow force acting on the spool is not affected by the outlet pressure.

B. THE EFFECT OF INLET VELOCITY
The effect of the inlet velocity on the radial force was studied at five values: 2.5 m/s, 5 m/s, 7.5 m/s, 10 m/s, and 15 m/s, with the orifice opening kept steady at 0.5 mm and the outlet pressure at 5 MPa. Fig. 9 shows five curves, one for each of these velocities, for the pressure difference around the annular orifice between P up and P down . There is a noticeable  increase in the pressure difference as the inlet flow velocity rises, and that value is proportional to the square of inlet flow velocity. By means of the following equations (4) and (5), this could be explained in terms of mass and energy conservation [28], [29]: where Q in is the inlet flow rate, given in m 3 /s; A in is the area of inlet port P, in m 2 ; and A o is the area of the orifice, in m 2 . Accordingly, p can be expressed as  The radial flow forces under the five distinct inlet velocities were calculated on the basis of formula (1) from the data presented in Fig. 9, and Fig. 10 presents the five corresponding curves for radial flow force difference F r , which were obtained by using absolute radial flow forces to subtract out the minimum value for each condition. Table 2 presents the amplitudes of radial flow force F ra for the various inlet velocities, and the trend of increase in F ra with inlet velocity is shown in Fig. 11. For our calculations, we used the r i ratio, VOLUME 8, 2020  defined as the ratio of radial flow force amplitude F ra to the square of the inlet flow velocity. This is calculated thus: where n refers to the serial number listed in Table 2. The four values of r i presented in Fig. 11 are calculated for n = 1 to n = 4. That the r i curve for the value of 1 is close to a horizontal straight line illustrates the fact that the amplitude of radial flow force is proportional to the square of inlet flow velocity, which can be verified by substituting formula (5) into formula (1). Fig. 12 shows radial flow forces for four orifice opening values -namely, 1 mm, 2 mm, 3 mm, and 4 mm -when the outlet pressure is 5 MPa and inlet velocity is 10 m/s. The curves demonstrate that the radial flow force decreases as the orifice opening grows. The radial flow force exhibits a reciprocal relationship with the orifice opening, as attested by (7), which is derived from the above-mentioned substitution.

C. THE EFFECT OF THE ORIFICE OPENING
The amplitude values of the four curves shown in Fig. 12 as the opening expands from 1 mm to 4 mm are 6.08 N, 7.03 N, 9.17 N, and 9.85 N.

D. RADIAL FORCE
To study the radial force acting on the spool in response to the asymmetrical radial flow force at the annular orifice, one can calculate the net radial force by projecting the radial flow force in the Y and Z direction and summing in the same direction to find the square roots. The components of the net radial force in the Y and Z direction can be calculated as, respectively, where F rY is the Y component of the net radial force and F rZ is the Z component. Net radial force F r acting on the spool at the orifice is the square root of the F rY and F rZ values, per formula (10), and the angle of the radial force to the positive direction of the z-axis α can be calculated via (11): and As the radial flow force distributions are not circumferentially symmetrical at the annular orifice, the distribution trend is plotted to show how the radial flow force varies around the orifice. Fig. 13 presents the distribution of radial flow force around the orifice for one sample condition, to illustrate its asymmetry. For this condition, the net radial force and angular   Table 3.
The values in Table 3 indicate that the radial force increases as the inlet flow velocity grows, when the opening and outlet pressure are constant. This is because of the asymmetric distribution of radial flow force. Also, the amplitude of that force grows with the inlet velocity, as shown in Figs. 10 and 11. The net radial force is greater when the opening is equal to or larger than 1 mm, relative to that with a 0.5 mm opening at the same inlet velocity and outlet pressure. The magnitude of the net radial force changes with the opening size, which is also affected by the amplitude and symmetry of the radial flow force. The net radial force manifests itself near 0 • and is oriented in the negative direction on the z-axis when the opening size is 0.5 mm. When the opening is at least 1 mm, the net radial force appears in the lower-left quadrant with reference to Fig. 13. Therefore, one can conclude that the direction of the total radial force changes with the opening of the orifice, and the direction moves from the negative Z direction to negative Y.
The results presented above illustrate that the radial force induced when fluid flows through the orifice must not be neglected for large openings or cases of high inlet velocity, because the net radial force could lead to spool eccentricity or tilting of the bore centre line, which generates extra friction resistance in the process of sliding.

V. CONCLUSION
Valuable results emerged from this research into 1) the radial flow force's distribution and its effects around the annular orifice of the main stage of a 2D servo valve and 2) the net radial force acting on the spool in consequence of the fluid flows. Firstly, the results indicate that the radial flow force arises from the asymmetric fluid cavity, while outlet pressure had no effect on that force's distribution. Our work demonstrated that the radial flow force distribution is affected by the inlet flow velocity and the opening of the orifice, with the amplitude of the radial flow force increasing linearly with the square of inlet velocity and showing an inverse relationship with the opening size. The net radial force acting on the spool was verified as not representing mutual cancellation of the forces, and it grew with the inlet flow velocity and the orifice size. These results can inform not only analysis of the radial force operating at the main stage of a 2D servo valve but also work with traditional slide valves with a conical side surface.
Further research should focus on experiments to validate the numerical results presented in this paper. We would recommend modifying the model of the valve's dynamics so as to account for the effect of the radial flow force. With a model modified thus, new structures for slide valves could be designed, enabling better compensation for both axial and radial flow forces. Thereby, better valve performance should be produced.