#### Abstract

As an actuator control element of the electrohydraulic system in aerospace, airplanes, and other equipment, the spool valve is prone to the “hydraulic lock” problem, which may cause major accidents of these electrohydraulic systems. The grooves engraved on the spool can effectively eliminate this problem by migrating the uneven pressure distribution in the clearance film between the sleeve and the spool. The effect of migrating uneven pressure distribution depends on the groove parameters. This paper proposed an accurate mathematical model with rectangular grooves to investigate the effect. Unlike previous mathematical models based on the Reynolds equation, the proposed model is based on the Navier–Stokes equation, which is more valid in the range where the clearance film thickness is much less than the groove depth. Meanwhile, the mathematical model takes the distributions, width, depth, and the number of grooves into consideration. Then, the proposed mathematical model was used to investigate the effects of mitigating the uneven pressure distribution under various parameters of grooves. To verify the accuracy of the mathematical model, the volume flow leakage in the clearance film obtained by the proposed model and the Reynolds equation, respectively, is compared with that obtained by the experimental test. The comparison results indicate that the results obtained by the Reynolds equation could reach a maximum of 14.75% different from the experimental results, while the results obtained by the NS equation are only 5.57% different under the same conditions, implying that the mathematical model derived from the NS equation is more accurate.

#### 1. Introduction

As the amplifier stage of an electrohydraulic system, the spool valve provides control over the direction and flow of the fluid, which is widely used in various hydraulic actuators that can be utilized to control the aircraft rudder surface deflexion, control the drilling steering, control the turning of a ship, and so on. Therefore, the spool valve affects the hydraulic actuator’s position control accuracy, speed control accuracy, and force control accuracy. However, the spool-type valves are susceptible to a unique problem known as hydraulic lock due to the uneven pressure distribution in the clearance film between the spool and the sleeve. The uneven pressure distribution around the spool creates a lateral force which causes the spool to tilt at a small angle. Then, the lateral force increases with the tiny angle and makes the tilted angle greater so that the spool contacts with the inner wall of the sleeve, resulting in the spool locking in the sleeve. The spool locking phenomena cause the actuator of hydraulic equipment to fail and even cause safety accidents, as shown in Figure 1. In theory, the locking problem could be avoided when the clearance exactly parallels with the sleeve’s walls. However, in practice, the perfectly parallel walls between spools and sleeves are not obtainable due to the limitations of tooling machines. Fortunately, some circumferential grooves applied on the spool surfaces can be used to decrease the lateral force to limit the effects of locking phenomena.

The first study about limiting the hydraulic locking phenomena was conducted by Sweeney [1], who presented that some grooves along the spool axial direction can mitigate uneven pressure distributions in the clearance film. Since then, several researchers [2–10] have utilized the Reynolds equation to investigate the lubricant characteristics of the clearance film of the spool valve. For instance, the pressure in each groove and the leakage in the clearance film were calculated by Milani [10] based on the Reynolds equation. Since the Reynolds equation is derived from the NS equation, Dong et al. [11] compared the difference of pressure distribution between the Reynolds equation and the NS equation and found that the difference exceeds 20% when the clearance film is greater than 0.413 . Furthermore, Guardino et al. [12] investigated the difference in various values of the roughness amplitude to the clearance film ratio and pointed out that the difference increases with the ratio. In the following years, some research results [13–18] used in hydrostatic gas-lubricated tribological components obtained the same results as those in reference [12]. Simultaneously, the research methods based on the NS equation are applicable to the study of the clearance film lubrication between the sleeve and the spool as the depth of the groove to the thickness of the clearance film ratio is greater than ten [19, 20]. Therefore, Hong and Kim [20] used computational fluid dynamics (CFD) based on the NS equation to compare spiral grooves’ migrating uneven pressure distribution effects and traditional grooves. The uneven pressure distribution was simulated for different inlet pressure conditions [21]. It was found that the uneven pressure increased linearly with the inlet pressure. Unfortunately, the study [21] did not investigate the effect of grooves on eliminating uneven pressure. Meanwhile, CFD was widely utilized to study the effect of different types of grooves on improving load-carrying capacity and film stiffness in gas face seals [22–25]. Load-carrying capacity and film stiffness in the gas face seal are related to the type of grooves and the geometric values of grooves. However, load-carrying capacity and leakage rate are conflicted. In other words, the improvement of load-carrying capacity increases the leakage rate, which must be as small as possible for the gas face seal. Aiming to take into account load-carrying capacity and volume flow leakage, Wang et al. [26] firstly established an accurate mathematical model of grooves and then utilized this mathematical model to optimize the groove shapes using the multiobjective optimization approach, which is very mature in the present time. Thus, the accurate mathematical model of grooves is the most important for the multiobjective optimization of groove shapes. Unfortunately, the current mathematical models of the clearance film pressure distribution between the sleeve and the spool are based on the Reynolds equation. The difference of pressure distribution in the spool valve between the Reynolds equation and the NS equation increases with width, depth, and number of grooves [27].

Therefore, an accurate mathematical model of clearance film pressure distribution in the spool valve is established based on the NS equation in cylindrical coordinates in this paper. Meanwhile, this accurate mathematical model takes into account the distribution, width, depth, and the number of grooves and is used to investigate the lateral force and the volume flow leakage under various groove geometric dimensions. To verify the mathematical model, an experiment is tested to compare with the theoretical results obtained by the NS equation and the Reynolds equation, respectively.

#### 2. Description of the Spool Valve Structure

The spool valves are utilized to amplify the hydraulic energy in the servo valve, as shown in Figure 2(a), and to control the flow’s direction and flow in the spool valve system, as shown in Figure 2(b). Figure 2(c) illustrates that the spool valve consists of the sleeve and the spool, and a spool usually has several lands. For the convenience of calculation, the mathematical model of one land is first established. Thus, the geometry of the spool can be simplified as Figure 2(d). The simplified spool with radius *r*_{sp} and length 2*l* is nested in the sleeve with the radius *r*_{sl}. The spool is stationary with a tilted angle in the sleeve.

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The classical groove applied to the spool mostly forms a circular groove with a rectangular cross-sectional area. The parameters of the groove consist of the distance between the groove and the spool edge, the distance *l*_{3} between grooves, the width *l*_{2} of groove, and the depth *h*_{z} of the groove, as shown in Figure 2(d). Supposing that the number of the grooves is *n*, the length of the spool can be written as

The geometric dimensions of the spool and the groove are tabulated in Table 1. The length of spool *l*, the radius of the spool *r*_{sp}, the inner radius of the sleeve *r*_{sl}, and are all constant, which do not affect the lubricant characteristics of the clearance film between the spool and the sleeve. Besides, the aspect ratio *K* represents the groove depth *h*_{z} to the groove width *l*_{2} ratio, the width of the spool *l*_{2}, the distance between grooves *l*_{3}, and the tilted angle which are variable geometric dimensions of the spool, which need to investigate the effect on the lubricant characteristics.

#### 3. Mathematical Model

As shown in Figure 2(d), taking into account the cylindrical shape of the spool and the sleeve, the NS equations in cylindrical coordinates are written aswhere

The sleeve and the spool are both assumed to be stationary. In addition, the tilted angle of the spool is small enough that the flow in the clearance film between the spool and sleeve moves approximately along the *z*-direction. The terms for velocity in the NS equation thereby can be written asWhen (4) is substituted into (2), and the second-order differential terms in (2) are reduced, (2) can be written as

The integral general solution of fluid velocity can be listed as follows:where is the fluid’s dynamic viscosity, *r* is the radius with *O* as the center, and *C*_{1} and *C*_{2} are both constants.

According to the assumption above, equals zero when *r* in (6) equals the radius *r*_{sl}. When *r* equals *r*_{z} which is the distance from the center *O* to the wall of the spool, *r*_{z} equals zero since the spool is stationary. *r*_{z} is a variable parameter, the value of which is related to the center distance and the rotary angle , that needs to be calculated using the longitudinal section illustrated in Figure 3. Due to the very small tilted angle , the longitudinal section of the spool is approximately circular whose center is *O*_{sp}, and the radius is . As shown in Figure 3, the eccentricity direction is assumed to coincide with the angles increasing in the clockwise direction. Then, the analytical expression for *r*_{z} can be deduced as follows:

Introducing the boundary conditions and putting (7) into (6) by linearizing to , the fluid velocity in the clearance film becomeswhere the term is eliminated in the calculation, and thus, the effect on the calculation of linearization above is ignored.

The differential element of volume flow leakage in the clearance film can be written aswhere the range of *r* is from *r*_{z} to *r*_{sl}. Integrating (9), *Q* can be expressed as

Introducing (8) into (10), the integration for *Q* is solved as follows:

According to (11), the analytical expression of the pressure gradient along the *z*-axis becomes

Since the range of spools in the *z*-axis is from −*l* to *l*, the pressure distribution in the clearance film can be written in (13) by solving a definite integral pressure gradient:where *C*_{3} is a constant, and is a function with *z* as its argument. According to an integral part of (12), is expressed as

For the pressure distribution at both ends of the spool, the first boundary conditions are *P* = *P*_{2} and *z* = *l*, and the second boundary conditions are *P* = *P*_{1} and *z* = −*l*. Thus, owing to the continuity of fluid, the constant *C*_{3} can be written as

The volume flow leakage in the clearance film is written as

Thus, from equation (13) to (15), we can obtain the following:

The value of *h*_{z} in (14) is related to the *z*-axis coordinate value and the number of grooves. When the number is even, the value of *h*_{z} becomeswhere *m* is the number of grooves, and the value can be set as 1, 2, 3, , *n*/2.

When the number is odd, the value of becomeswhere the value of *m* can be set as 1, 2, 3, , (*n* + 1)/2, .

Equation (17) presents the pressure distribution in the clearance film between the spool and the sleeve, which can be used to calculate the lateral force by integrating the pressure *P* in the clearance film, as shown in (20). The decrease rate of the absolute value of the lateral force reflects the effect of eliminating the uneven pressure distribution around the spool [21]. The smaller the lateral force, the better the effect of eliminating the uneven pressure distribution:

#### 4. Simulation Results

##### 4.1. The Optimal Distribution of the Grooves

The optimal distribution of the grooves needs to be confirmed first. For the sake of investigating the effect of parameter *l*_{3} on the pressure distribution in the clearance film, several cases are compared through calculating the pressure distribution and the lateral force based on the geometric dimensions tabulated in Table 1. Except for the parameter *l*_{3}, each case utilizes the same geometric dimensions, such as width (*l*_{2} = 0.8 mm), depth (*K* = 0.25), and the number of grooves. Meanwhile, each case shares the same properties of the working fluid, as shown in Table 2, and utilizes different pressure conditions, as shown in Table 3. With the decrease of *l*_{3}, the grooves are clustered to the end edge of the spool (*l*_{3} = 0.1 mm). When *l*_{3} equals 1.6 mm, the grooves are evenly distributed.

Figure 4(b) illustrates that the lateral force has a minimum value in the case where the grooves are evenly distributed. And the equidistribution grooves have the same effect on the pressure distribution in the case of different pressure conditions and tilted angles, as shown in Figures 4(c) and 4(d).

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Figures 4(c) and 4(d) also illustrate that the decreased slope of the lateral force with an increment of *l*_{3} under a tilted angle, *α* = 0.0188°, is lower than that under the tilted angle *α* = 0.0288°. This is because the effect of grooves decreases with the reduction of tilted angle . Nevertheless, compared with the lateral force at *l*_{3} = 0.1 mm, the lateral force at *l*_{3} = 1.6 mm is reduced by 16.9%, 14.9%, and 14.8% at pressure 5 MPa, 7 MPa, and 9 MPa, respectively. Same as the value of the lateral force, the uniformly distributed grooves provide a more effective lubricant characterized along the circumference, mitigating the unbalance pressure distribution. Figure 4(a) supports the effect, which is very approximate to the numerical results in reference [27]. Therefore, compared with other distributions of grooves, the even distribution groove is optimal.

##### 4.2. The Effect of the Cross-Sectional of Groove on Lateral Force

Besides the distribution of grooves, the geometric dimensions also affect the value of the lateral force, such as the groove depth *h*_{z} and the groove width *l*_{2} [27]. Therefore, it is essential to investigate the effect of these two parameters on the lateral force. As shown in Figures 5–7, the lateral force decreases with the increase in the width of the groove for the same value of *K* in cases where the number of grooves is 2, 4, 6, and 8, respectively. Meanwhile, the lateral force also decreases with the increment of the aspect ratio *K* for the same value of *l*_{2}. This is because the lateral force is related to the cross-sectional area which is proportional to the width *l*_{2} and *K*, as shown in Figure 8. In other words, the increase in the value of the cross-sectional area indicates a decrease in the lateral force value. As a result, the increase in cross-sectional helps migrate the uneven pressure distribution surrounding the spool.

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In addition, different increasing rates of cross-sectional area of the groove lead to different decreased slopes of the lateral force. For instance, as shown in Figure 5(a), the greater value of *K* leads to the greater increase rate of the cross-sectional area, which causes the greater decreased slope of the lateral force for the same increased rate of *l*_{2}. Similarly, different groove numbers result in different decreased slopes of the lateral force. As shown in Figure 9, for the same value of *K* and the same increased rate of the groove width *l*_{2}, the decreased slope of the lateral force increases with the increase in the groove number in the case where the pressure *P*_{1} = 7 MPa and the tilted angle = 0.288°. For example, in the case where *K* = 0.25, along with the same increased rate of *l*_{2}, the reduction rate of the lateral force is 3.35%, 5.91%, and 17.49% when the number of the grooves is eight compared with which is 1.73%, 2.89%, and 4.4% when the number of the groove is 4. This conclusion also applies to the reduction rates of the lateral force in cases of *P*_{1} = 5 MPa and 9 MPa, respectively, and the tilted angle = 0.188°, as tabulated in Tables 4–6.

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According to the theoretical results above, the lateral force decreases with the increase in the groove number. However, the volume flow leakage increases with the increase in the groove number, which is illustrated in Figure 10. Meanwhile, the volume flow leakage increases with the increase of *P*_{1} since the volume flow leakage is positively proportional to the differential pressure according to (16). As the sole driving energy of the spool, the differential pressure, as shown in (21), decreases when the volume flow leakage is too much. In other words, too much volume flow leakage causes the reduction of the driving force of the spool, to reduce the dynamic characteristic of the valve:

In addition, the strength of the spool reduces with the increase in groove depth. Therefore, the width of the groove, the depth of the groove, the number of the grooves, the leakage volume flow, and the strength of the spool need to be comprehensively considered when optimizing the parameters of the groove.

#### 5. Experimental Verification

As for the second or third stage of the servo valve or other valves, the spool usually has more than one land. To conclude more closely to fit the actual situation, the second stage spool of the servo valve with four lands is investigated. Figure 11 shows the spool valve, and the spools with four lands are named as *S*_{1}, *S*_{2}, *S*_{3}, and *S*_{4}. As shown in Figure 11, *P*_{1} and *P*_{2} are the relatively high-pressure inlet and the relatively low-pressure outlet, respectively, and *P*_{A} and *P*_{B} are the loading outlet of the servo valve.

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The hydraulic diagram of the test bench is shown in Figure 12. The two overflow valves are used to change the supply oil pressure. The solenoid valve 4 is normally open. When the solenoid valves 1 and 2 are both turned on, and the solenoid valve 3 is turned off, the volume flow leakage between the land *S*_{2} and the sleeve can be tested by flowmeter. Conversely, when the solenoid valves 1 and 3 are both turned on, and the solenoid valve 2 is turned off, the volume flow leakage between the land *S*_{3} and the sleeve can be tested by flowmeter.

To verify the mathematical model derived above, the volume flow leakage rate between the spool and the sleeve is tested by the equipment setup illustrated in Figure 13. The test system consists of the oil supply system, the spool valve drive system, and the data acquisition system. In Figure 13(b), the spool valve drive system is composed of the stepper motor, and the spool valve is preinstalled and tilted at an angle . The stepper motor propels the spool towards the positive direction of the *z*-axis until the port *A* is fully open. Meanwhile, port *A* is connected to the relatively high port, and port *B* is connected to the relatively low-pressure port. Then, the volume flow leakage between the land *S*_{2} and the sleeve is measured by the flowmeter, that is shown in Figure 13(c). Conversely, when the spool was pulled in the negative direction of the *z*-axis, the volume flow leakage rate between the land *S*_{3} and the sleeve was measured and recorded. Figure 13(d) presents the oil circuit connection photo at the bottom of the spool valve. Thus, *S*_{2}, *S*_{3}, and the sleeve can be obtained at different differential pressure values derived from the impairment between *P*_{1} and *P*_{2}.

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The oil supply system can provide different constant values of pressure, where the value of return oil is 0.3 MPa so that the value of *P*_{2} is set as 0.3 MPa in the mathematical model. The boundary conditions of the land *S*_{2} are shown in Figure 14. To keep the boundary conditions consistent with the theoretical calculation, the relatively high-pressure P_{1} in the experiment is set as 5.3 MPa, 7.3 MPa, and 9.3 MPa, respectively, taking the spool valve used in hydraulic dives and control, where working pressure often reaches 35 MPa. The relatively high-pressure P_{1} which equals 35.3 MPa is considered both in the simulation and experiment. Then, the differential pressure *P*_{d} in the experiment is the same as that in the theoretical calculation. Simultaneously, the number of the groove and the tilted angles of the spool are both taken the same values as the values of the theoretical calculation, as tabulated in Table 1.

According to the boundary conditions above, the volume flow leakage rate of the theoretical results and experimental results is compared in Figure 14. The theoretical results and the experimental results both present that the volume flow leakage rate increases with the increase in the number of grooves and the differential pressure. In addition, since under the same inner diameter of the sleeve, the greater the tilted angle of the spool, the smaller the clearance film between the spool and the sleeve, the volume flow leakage rate of the theoretical results decreases with the increase in titled angle, which is just same as that of the experimental results.

Meanwhile, Figure 15 shows that the volume flow leakage of the theoretical results is smaller than that of the experimental results in the cases where the differential pressure is 5 MPa and 7 MPa, but it is the opposite in the case where the differential pressure is 9 MPa and 35 MPa, respectively. With the increase of the differential pressure, the linearity of the theoretical volume flow leakage value is better than that of the experiment, which is more clearly shown in Figure 15. This is because the dynamic viscosity of the fluid increases with the increase in differential pressure, which is ignored in the theoretical model. Nevertheless, the theoretical results are close to the experimental results, which verify the mathematical model derived above.

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As shown in Figure 15, the “NS equation” and “Reynolds equation” represent the results obtained by the NS equation and the Reynolds equation from reference [21], respectively. The comparison results show that the volume flow leakage obtained by the NS equation is closer to the experimental results than that obtained by the Reynolds equation. For instance, the maximum percentage of the difference between the results obtained from the Reynolds equation and experimental results is 14.75% in the case where the differential pressure *P*_{d} is 5 MPa, the groove number *n* is 6, and the tilted angle is 0.0188°. Under the same condition, the percentage of the difference between the results obtained from the NS equation and experimental results is 5.57%. That is to say, the mathematical model of the spool valve clearance film proposed in this paper is more accurate than the models derived from the Reynolds equation.

#### 6. Conclusions

This study establishes an accurate mathematical model of the pressure distribution in the clearance film between the sleeve and the spool with rectangular grooves. This model was derived from the NS equation in cylindrical coordinates and can be used to calculate the uneven pressure distribution, lateral force, and volume flow leakage for the incompressible, isothermal, and Newtonian flow in the clearance film between the sleeve and the spool in different configurations such as geometric dimensions of the spool width, depth, and number. In addition, the values of the volume flow leakage obtained by the mathematical model are compared with that obtained by the experiment under various boundary conditions. The theoretical results and the experimental results indicate conclusions as follows:(1)Compared with the experimental results, the mathematical model derived from the NS equation could more accurately calculate the lubricant characteristics of the fluid in the clearance film between the sleeve and the spool with rectangular grooves than the model derived from the Reynolds equation in the case of various parameters. However, the effect of the dynamic viscosity changes due to the differential pressure on the uneven pressure distribution required in the future.(2)The uniformly distributed grooves provide a more effective lubricant characterized along the circumference, mitigating the unbalance pressure distribution. Furthermore, the theoretical results suggested that the wider, deeper, and more number of grooves could more effectively mitigate the unbalanced pressure distribution. In other words, an increase in the cross-sectional area of the groove could effectively decrease the lateral force acting around the spool.

Although the increase in cross-sectional area of the groove could reduce the lateral force, it also degrades the performance of the spool valve. For instance, the increase in volume flow leakage owing to the increment of the number of grooves reduces the driving energy of the spool. Meanwhile, the increment of the depth of the groove reduces the structural strength of the spool. In addition, the mathematical model established in this paper neglects the variation of dynamic viscosity with pressure although the theoretical results are close to the experimental results.

#### Data Availability

The (groove_with_L2_change_n_8.m) data used to support the findings of this study are included within the article. The (groove_with_L2_change_n_8.m) data present the lateral force in the case where P1 = 7 MPa, = 0.288 degree, K = 1, and *l*_{2} = 0.2 mm. The data can be run in MATLAB R2016b.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This work was supported by the National Science Foundation of China (Grant No. U20B2029), National Natural Science Foundation of China (Grant No. 52174005), Natural Science Research Program of the Department of Science and Technology of Shaanxi Provincial (Grant No. 2020JQ-776), (Grant No. 2021KW-33), and (Grant No. 2022JM-305).