**1. Introduction**

Surface microtexturing has gradually attracted attention due to its good hydrophobicity and drag reduction properties, which has become an effective way to improve the performance of new products and reduce energy consumption. Since Hamilton and Halavit [1] first proposed to process the microstructure on the surface of the friction pair to obtain the dynamic pressure effect, many scholars have carried out research in this area. The effects of texture morphology, geometric parameters, texture distribution and environmental factors on the lubricating performance were investigated by simulation and experiment. Rahmani et al. [2] combined the Reynolds equation and surface discontinuity of the hydrostatic bearing to establish a model and studied the influence of various forms of surface texture on the friction and carrying capacity of the hydrostatic bearing and believed that the form of texture has a significant impact on the bearing lubrication performance; waist triangle, right triangle and rectangular microtextures significantly improved the lubricating performance. Sinano˘glu et al., used experimental methods to verify that the

**Citation:** Shang, Y.; Cheng, K.; Bai, Q.; Chen, S. Drag Reduction Analysis of the Hydrostatic Bearing with Surface Micro Textures. *Appl. Sci.* **2022**, *12*, 10831. https://doi.org/ 10.3390/app122110831

Academic Editors: Josep Maria Bergadà and Gabriel Bugeda Castelltort

Received: 9 August 2022 Accepted: 17 October 2022 Published: 26 October 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

trapezoid has a higher load-carrying capacity than the zigzag texture, and the maximum pressure value is reduced by 36% compared to the smooth journal. In terms of pit-like textures, Tala-Ighil et al. [3] analyzed the effects of spherical, cylindrical and parallelepiped textures on the minimum oil film thickness, maximum pressure, axial flow velocity and friction force on the shaft diameter surface under the condition of stable flow field by using the finite element method, and the cylindrical texture performance improved more. Wang [4] found through experiments that with a reasonable design of geometric shapes and distributions, micropits can obtain nearly 2.5 times the carrying capacity of smooth surfaces.

Rahmani's [2] research shows that texture height has a greater impact on tribological properties than texture area ratio. Through experimental comparison, Sinano˘glu [5] found that reducing the texture height and inclination would reduce the LCC. Cupillard [6] applied numerical methods to study the effect of microgrooves on the axial direction of the bearing pad wall on the lubricity of radial hydrostatic bearings when cavitation effects were considered. It shows that the microgrooves can increase the minimum oil film thickness and reduce the frictional resistance under the condition of small depth and low load (eccentricity *ε* < 0.15); under the condition of large depth and high load (*ε* > 0.15), the microgrooves can reduce the friction force, but simultaneously, the thickness of the oil film is reduced. In terms of thrust hydrostatic bearing research, Wang [4] used the load capacity/friction coefficient transition point to measure the performance of the bearing, and studied the effects of the ratio of the area of the crater, the depth of the crater and the diameter of the crater in the transition point of the LCC and obtained the best result. Zhang [7] optimized the geometrical parameters of the fan-shaped straight groove texture on the bearing surface and considered that the microtexture had different optimal parameters under different constraints and objectives. He analyzed the effect of groove texture on the fluid thrust lubrication mechanism of thrust washers and found that the width and number of grooves have optimal values for load carrying capacity.

Regarding the location of texture distribution, Cupillard [6] simulation found that the microtexture model arranged in the maximum pressure region can reduce friction and oil film thickness compared with the smooth wall model. Liu Hongbin's numerical study found that there is a reasonable density and distribution for the oil film LCC. Brizmer [8,9] found in the micropit texture research that the global texture model cannot effectively improve the LCC, but it is suitable for mechanical seals.

Hamilton [1] proposed the "lubrication effect of microbumps", which believes that the microbumps on the surface hinder the movement of the fluid. When the fluid flows through the microbumps, the pressure rises rapidly and cavitation causes the pressure asymmetry, resulting in additional LCC. Cupillard's [10,11] numerical analysis of textured slider bearings found a "reflow phenomenon". When the texture depth increases to a certain value, backflow will occur, and the backflow will reduce the pressure gradient and LCC. Lo [12] found that the pit-textured surface lubricating fluid penetrates into the surrounding area to reduce friction and wear. Wang experimentally verified this phenomenon and summarized it as "secondary lubrication effect". Fowell [4] discovered the phenomenon of "inlet suction" when studying low-convergence textured wall bearings: unlike the classical shear force that entrains the lubricating oil to form hydrodynamic pressure, the phenomenon is that the textured area at the bearing inlet produces a lower pressure than the outside air. The high pressure causes the lubricating oil to be squeezed into the bearing, thereby increasing the LCC of the bearing. Tauviqirrahman et al. [13] believed that the surface microstructure affects the slip length of the fluid–solid surface, the journal bearings lubricated by Newtonian fluid and non-Newtonian fluid are compared, and it is concluded that non-Newtonian fluid is more effective in improving the load-carrying capacity in the case of high eccentricity.

Numerous studies have shown that applying a reasonable surface texture can lead to better tribological performance than smooth surfaces, but some issues still need to be investigated [14]. Whether there are some special textures that can achieve the best results performance in any situation; how different calculation models used affect the accuracy of the calculation results.

### **2. Analysis of Smooth-Walled Radial Bearing Ignoring Cavitation Effects**

### *2.1. Governing Equations*

In the process of high-speed rotation of hydrostatic bearings, the oil film of journal bearings will form negative pressure under the action of oil wedge. This cavitation phenomenon has been considered by a large number of scholars, and the most common is the half-Sommerfeld method. However, it is more practical to explain this phenomenon through gas–liquid two-phase flow [13,15]. Therefore, the conservation of momentum and energy are formulated as follows:

Mass conservation equation:

$$
\nabla \cdot \vec{\nabla} = 0 \tag{1}
$$

Momentum equation:

$$
\rho \frac{D\vec{V}}{Dt} = -\nabla p + \rho \vec{\text{g}} + \mu \nabla^2 \vec{V} \tag{2}
$$

We can see that the momentum equation is related to the density, viscosity and the volume velocity of the micro element, extending Equation (2):

$$\frac{\partial}{\partial \mathbf{x}\_{j}} (\rho u\_{i} u\_{j}) = -\frac{\partial p}{\partial \mathbf{x}\_{i}} + \frac{\partial}{\partial \mathbf{x}\_{j}} \left( \mu \left( \frac{\partial u\_{i}}{\partial \mathbf{x}\_{j}} + \frac{\partial u\_{j}}{\partial \mathbf{x}\_{i}} \right) \right) + \frac{\partial}{\partial \mathbf{x}\_{j}} \left( -\rho \overline{u\_{i}^{\prime} u\_{j}^{\prime}} \right) \tag{3}$$

Energy equation:

$$\frac{\partial}{\partial t}(\rho E) + \nabla \cdot [\vec{v}(\rho E + p)] = \nabla \cdot \left[k\_{eff} \nabla T\right] + \mathcal{S}\_h \tag{4}$$

The two-dimensional schematic diagram of a radial hydrostatic bearing, shown in Figure 1.

**Figure 1.** Schematic diagram of a radial hydrostatic bearing.

The main parameters of oil are: density 810 kg/m<sup>3</sup> ; specific heat capacity 2000 J/(kg · K); thermal conductivity 0.37 W/(m · K); dynamic viscosity of lubricating oil is 3.85 <sup>×</sup> <sup>10</sup>−<sup>3</sup> Pa · s (temperature *<sup>T</sup>* = 293 K). Analysis parameters are shown in Table 1.

**Table 1.** Calculation parameters for radial hydrostatic bearing.


According to the Reynolds number calculation formula:

$$Re = \frac{\rho v d}{\mu} \tag{5}$$

where *d* is length scale, and *v* is reference velocity. Based on the maximum oil film thickness, let *d* = 2*h*max, the Reynolds number calculated as about 75, which is much lower than the range of 1000 to 2000, which can generate turbulent flow. Therefore, this problem can be calculated according to the laminar flow theory, and the obtained pressure distribution in the oil film is shown in Figure 2.

**Figure 2.** Pressure distribution of oil film on the journal hydrostatic bearing: oil pressure (MPa).

Based on the maximum oil film thickness, the Reynolds number is calculated as about 75, which is much lower than the range of 1000 to 2000, which can generate a turbulent flow. Therefore, this problem can be calculated according to the laminar flow theory, and the obtained pressure distribution in the oil film is shown in Figure 2.

In order to verify mesh independence, three different proportions of mesh number are compared with the FEM method. As shown in Table 2.


**Table 2.** Mesh independence verification.

It can be seen that without considering the cavitation effect, the minimum pressure may be negative pressure, which is obviously inconsistent with the actual situation. Therefore, the cavitation effect must be considered to obtain reasonable results. At the same time, the number of the mesh has little influence on the calculation results, which proves the independence of the mesh.

The process of generating, compressing, expanding and collapsing of tiny bubbles in a liquid is called cavitation. Several scholars have made outstanding contributions to the mathematical theory of cavitation effects. It was first predicted by the physicist Reynolds in 1873 that the high-speed relative motion between the propeller and the water would create a vacuum cavity that affects the performance of the propeller during the travel of the ship. Later, S.W. Barnaby et al. [16]. However, cavitation is not the only disadvantage. Along with the cavitation process, the bubbles generated by the fluid and the moving wall reduce the macroscopic shear force during the movement. For example, supercavitation

technology is used to generate cavitation on the surface of the torpedo, so that the resistance of the torpedo can be reduced by 90% [17], among which the "full cavitation model" was studied by Singhal et al. It accounts for phase change, bubble dynamics, turbulent pressure fluctuations, and noncondensable gases. It is able to take into account multiphase flows or flows with multiphase transport of matter, the effect of sliding velocity between liquid and gas phases, as well as the thermal effects and compressibility of liquid and gas phases, establishing the Rayleigh–Plessset equation, also known as the Porisky equation, which is also a widely used cavitation model [18].

Since the negative pressure problem in the diverging wedge region needs to be solved in combination with the cavitation phenomenon, the bearing oil film model is divided into two simplified slider bearings, the convergent wedge and the diverging wedge, which are analyzed separately.

With the multiphase cavitation modeling approach, a basic two-phase cavitation model consists of using the standard viscous flow equations governing the transport of mixture phases. In cavitation, the liquid–vapor mass transfer equation has the following form:

$$\frac{\partial}{\partial t}(a\rho\_{\upsilon}) + \nabla \cdot (a\rho\_{\upsilon}\vec{v}\_{\upsilon}) = R\_{\varepsilon} - R\_{\varepsilon} \tag{6}$$

In the equations mentioned above, *t* is time, *α* is volume fraction, *ρ<sup>v</sup>* is vapor density, ~*v<sup>v</sup>* is vapor velocity, *R<sup>e</sup>* and *R<sup>c</sup>* are mass transfer source terms connect to the growth and collapse of the vapor bubbles. For different cavitation models, the definition of *α* or other assumptions are different, resulting in different calculation methods of *R<sup>e</sup>* and *Rc*. However, several cavitation models can be written as similar formulas with different calculation methods of coefficients *Cvap* and *Ccond*.

$$\begin{cases} \begin{array}{c} \mathcal{R}\_{\varepsilon} = \mathcal{C}\_{vap} \rho\_{v} \rho\_{l} \sqrt{\frac{2}{3} \frac{(p\_{v} - p)}{\rho\_{l}}}\\ \mathcal{R}\_{\varepsilon} = \mathcal{C}\_{cond} \rho\_{v} \rho\_{l} \sqrt{\frac{2}{3} \frac{(p - p\_{v})}{\rho\_{l}}} \end{array} \end{cases} \tag{7}$$

The saturation pressure *p<sup>v</sup>* is corrected by an estimation of the local values of the turbulent pressure fluctuations:

$$p\_{\upsilon} = p\_{\text{sat}} + \frac{1}{2}(0.39\rho k) \tag{8}$$

where *k* is kinetic energy.

### *2.2. Method Selection for Cavitation Effect Calculation*

To account for cavitation in oil film CFD calculations, a multiphase flow model needs to be enabled. There are three commonly used multiphase flow models, namely the VOF model, the mixture model and the Eulerian model, and their characteristics are as follows.


For the bearing oil film model where the gas phase and liquid phase are mixed and the gas phase volume ratio is much greater than 10%, the VOF model mainly used to calculate the stratified flow or free surface flow is obviously not suitable. Therefore, the calculation model needs to compare the mixture model and the Eulerian model. Pull the model to make trade-offs. The selection of the mixture model and Eulerian model is mainly based on the following principles.


For the bearing oil film model with micro-texture on the wall surface, the flow field is more complicated. In order to improve the stability of the calculation, the mixture model is used to calculate the cavitation phenomenon.

### *2.3. Simulation of Smooth-Walled Radial Bearing Considering Cavitation Effects*

The pressure distribution of the smooth wall radial bearing calculate d in this paper is singular because cavitation effect is not considered. Here, the model considering cavitation effect is adopted, and the calculated pressure distribution is shown in Figure 3. The analysis results show that the results after considering the cavitation effect are more in line with the real situation, and the maximum pressure also increased.

**Figure 3.** Oil film pressure distribution considering cavitation effect: Oil pressure (MPa).

Figure 4 shows the cloud diagram of gas phase volume fraction distribution. The dark blue area indicates that the gas phase volume fraction is 0, that is, this area only contains oil, while other warm tone areas indicate that this area is a cavitation area. Obviously, the cavitation area is distributed in the bearing divergence wedge area. From the volume fraction of the gas phase in the cavitation area, it can be seen that the volume fraction of the gas phase increases gradually from the minimum oil film thickness to the maximum oil film thickness along the circumference. The area with less volume fraction of the gas phase corresponds to the initial stage of the cavitation. With the rotation of the journal, the cavitation is brought to the area with large oil wedge thickness and developed, and finally collapses and disappears in the area near the maximum oil film thickness.

The results show that the maximum velocity of the gas phase and the liquid phase are almost the same, and about 70% of the bubbles are distributed in units with a gas volume fraction of less than 9.71%.

### **3. Analysis of lUbrication Characteristics of Converging Wedge Slider Bearing Surface Texture**

### *3.1. Pressure Distribution Simulation of Non-Textured Convergent Wedge Slide Bearing*

The slider bearing is a simplification of the convergent area and the divergent area of the radial hydrostatic bearing, so that the calculation can be more refined. The size of the slider bearing is selected as shown in Figure 5.

**Figure 5.** Schematic diagram of slider bearing dimensions.

The lubricating oil is Mobil Velocite oil No 2, and the oil temperature is 293.15 K. The moving speed of the upper wall (Moving Wall) is *v* = 20 m/s, and the lower wall (state wall) is stationary. In order to ignore the influence of the two side walls facing the flow, the periodic boundary conditions are set in the analysis. The processing of periodic boundary conditions from the finite element analysis theory has the characteristics of accelerating the convergence, so it will not adversely affect the analysis results.

For the untextured smooth surface, applying CFD analysis, the flow velocity variation along the oil film thickness direction at the inlet position, the center position and the outlet position is obtained as shown in Figure 6.

In the figure, *x* = 5 mm is the entrance position, *x* = 0 mm is the center position, and *x* = −5 mm is the exit position.

**Figure 6.** Velocity distribution along the thickness of the oil film.

*3.2. Simulation of Convergence Wedge-Slide Bearing Pressure Distribution on the Streamwise and Transverse Textured Surfaces*

Figure 7 shows two microtexture models (streamwise and transverse) selected to analyze the effects of texture orientation on bearing performance.

**Figure 7.** Schematic diagram of slider bearing microtexture.

The micro-texture topography only considers the rectangular cross-section, and the micro-texture is uniformly distributed, with a micro-textured surface in the whole area on the state wall.

The dimensionless depth *h<sup>o</sup>* is defined as the ratio of the texture depth to the minimum oil film thickness. The minimum oil film thickness is set to 10 µm and the texture groove width is 50 µm. The effect of the dimensionless depth on the carrying capacity is calculated. As shown in Figure 8.

Figure 8 shows that the dimensionless LCC decreases with the increase of the dimensionless depth. The transverse texture has better LCC than the streamwise texture. With the increasing dimensionless texture depth, the dimensionless relative LCC of the transverse texture shows a decreasing trend, but it is not a rapid linear decrease like the streamwise texture, but is basically stable at around 0.78, that is, the transverse texture decreases. Compared with the smooth surface, the LCC of the structural model is reduced by about 22%.

**Figure 8.** Effects of dimensionless depth of microtexture on carrying capacity.

The variation of shear force on the fixed wall and moving wall of the streamwise and transverse microtextures with the dimensionless texture depth is obtained through analysis. Figure 9 shows the variation trend of the dimensionless shear force of the streamwise texture model.

**Figure 9.** Influence of dimensionless depth of microtexture on wall shear force in streamwise texture model.

Figure 10 shows that compared with the smooth model without texture, the wall shear force of the streamwise textured model is significantly reduced. The shear force of the moving wall is slightly lower than that of the fixed textured wall. When the dimensionless texture depth is 1, the dimensionless shear force of the upper plate wall is close to 0.8, that is, the drag reduction rate is about 20%.

Figure 10 shows the variation trend of dimensionless shear force of the transeverse texture model.

**Figure 10.** Influence of dimensionless depth of microtexture on wall shear force in transverse texture model.

Figure 10 shows that when the transverse micro-textured slider bearing is applied, the shear force between the moving wall and the fixed wall is quite different. The dimensionless depth is about 0.6 close to the minimum value, and then the shear force increases gradually. The dimensionless shear force of the moving wall is basically between 0.93 and 0.94. Usually, the drag reduction of hydrostatic bearings has no meaning for fixed walls, so the drag reduction characteristics reflected by the transverse texture are not obvious.

### *3.3. Lubrication Characteristic Analysis of Local Texture Model*

No matter what texture type is used, the above analysis shows that the LCC is lower than that of the untextured smooth wall model, which obviously does not meet the lubrication goals of improving LCC and reducing wall friction. Therefore, the loss caused by changing the full texture model needs to be considered.

First of all, it is determined that only suitable microtextures are produced in the required bearing parts, the length of the slider bearing is set as *L*, and two parameters *α* and *β* are defined to describe the size and entrance distance of the microtextures. The length of the microtexture is *l* = *α* × *L* and the entrance distance *d* = *β* × *L*. Combining *α* and *β*, the wall shear force and pressure distribution of the slider bearing were analyzed. Among them, the dimensionless shear force on the wall surface of the streamwise texture is shown in Table 3.


**Table 3.** Dimensionless shear forces for the streamwise local texture state wall.

The analysis shows that with the same value of *α*, the texture distribution on the inlet side can obtain a higher drag reduction rate.

Figure 11 calculates the pressure distribution of the smooth surface, transverse textured and streamwise textured slider bearings for setting *α* = 0.1, *β* = 0.9.

**Figure 11.** Influence of local texture on oil film pressure distribution.

It can be seen from the figure that the introduction of microtexture not only affects the small area of the texture distribution, but also affects the size and shape of the pressure distribution of the entire bearing. However, the maximum load-carrying capacity of a smooth bearing cannot be reached regardless of the distribution.

### **4. Analysis of Bearing Lubrication Characteristics under Cavitation Effect**

*4.1. Analysis of Lubrication Characteristics of Divergent Wedge Slider Bearing Based on Full Texture Model*

In the analysis of a convergent wedge texture model, CFD calculation can be completed only by using a pressure-based solver. For the divergent wedge texture model, the influence of cavitation should be considered.

Figure 12 shows the distribution of oil film pressure on the moving wall before and after considering the cavitation effect when the streamwise dimensionless texture depth is 0.5. The abscissa −5 mm point in the figure corresponds to the oil film inlet, and +5 mm corresponds to the oil film outlet. The figure on the left shows the situation without considering the cavitation effect. The oil film pressure distribution is similar to a parabola. When the *x*-axis coordinate is about −1 mm, there is a minimum pressure of −0.9 MPa (absolute pressure); the picture on the right shows the case considering the cavitation effect. On the moving wall, the oil film pressure in the middle of the wall is equal to the cavitation pressure. There is a sudden change in the pressure value at the inlet and outlet of the oil film, and the oil film pressure at the inlet. The strength first increases to 0.038 MPa and then decreases to the cavitation pressure, while the oil film pressure at the outlet surges from the cavitation pressure to about 0.082 MPa. The upper part of the left figure shows the pressure position of cavitation pressure, showing the huge difference between considering and not considering cavitation.

According to the previous analysis, the arrangement of the micro-texture at the inlet will lead to the reduction of LCC, while at the outlet, due to the consideration of cavitation effect, the negative pressure strength obtained without considering cavitation effect is corrected, which improves the LCC. Therefore, it can be imagined that if the cavitation effect is strengthened by using a micro-texture at the exit, the LCC may be improved. Figure 13 shows the corresponding dimensionless bearing capacities from the analysis of different dimensionless texture depths for a fully textured diverging wedge slider bearing.

**Figure 12.** Pressure distribution on moving wall before and after considering cavitation effect in streamwise texture.

**Figure 13.** The influence of the depth of streamwise texture on the dimensionless LCC.

Figure 13 shows that changing the depth of the texture has no obvious effect on improving the LCC, but it usually does not reduce the LCC. Figure 14 shows the dimensionless shear force obtained by changing the dimensionless texture depth analysis.

**Figure 14.** The effect of the depth of streamwise texture on the dimensionless shear force.

It can be seen from Figure 14 that increasing the texture depth is very effective in reducing the wall shear force, and its maximum drag reduction rate is likely to reach 22%.

In order to compare the two different directions of micro texture used in the previous convergent wedge, the oil film pressure analysis, dimensionless LCC and dimensionless shear force analysis of the divergent wedge with a transverse micro-texture are carried out. The characteristics of oil film pressure are almost the same as those in the streamwise micro-texture pressure distribution in Figure 12, but the LCC and shear force are quite different. Figure 15 shows the dimensionless LCC obtained by changing the dimensionless texture depth analysis.

**Figure 15.** Influence of transverse texture depth on dimensionless LCC.

Figure 15 shows that the LCC of the transverse texture model is greater than that of the smooth model. The dimensionless LCC first increases rapidly with increasing dimensionless texture depth, and when the dimensionless texture depth is 0.6, the maximum dimensionless LCC is 1.18, and then the dimensionless LCC decreases slowly.

The variation of transverse wall dimensionless shear force with dimensionless texture depth is shown in Figure 16.

**Figure 16.** Influence of transverse texture depth on dimensionless shear force.

Transverse texture can effectively reduce the dimensionless shear force on the fixed wall, that is, the shear force lower than that of the smooth wall model can be obtained. When the dimensionless texture depth is 0.6, the minimum dimensionless shear force can be obtained as about 0.85, and the corresponding drag reduction rate is about 12%. However, while reducing the shear force on the state wall, the transverse texture model increases the dimensionless shear force on the moving wall. For example, when the dimensionless texture depth is 0.6, the shear force increases by 6.5% compared with the smooth wall. For the divergent wedge, the increase of the LCC of the moving wall and the increase of the shear force are unfavorable factors for lubrication.

### *4.2. Simulation of Drag Reduction Effect of Local Texture Model of Divergent Wedge Slider*

Dimensionless parameters are defined in the same way as in the previous convergence wedge *α* and *β* to represent the length of the texture area and the distance from the exit, that is, when *β* at 0, the local texture is arranged at the exit. For the streamwise texture model, the dimensionless shear force of the moving wall obtained from the simulation analysis is consistent with *β*, and the relationship is shown in Figure 17.

**Figure 17.** The influence of the position of the streamwise texture on the dimensionless shear force on the moving wall.

Figure 17 shows that the texture arrangement at the oil film outlet of the divergent wedge can obtain a lower wall shear force, for example *α* = 0.3 and *β* at 0, the dimensionless wall shear force of the moving wall can be reduced to left and right, respectively. As the distance from the outlet increases, the shear force will increase, and the texture is arranged in an inappropriate area. Compared with the smooth surface, the shear force can be increased. For example, from the pressure distribution, if *α* ≤ 0.3, and if the texture is arranged at the entrance, it will be higher than the smooth wall by about 20%, because the texture area is in the pressure rise area of the oil film, and the corresponding texture boundary position just corresponds to the peak point.

By analyzing the hydrostatic block bearing with transverse local texture, the dimensionless shear force on the moving wall and *β*. The relationship is shown in Figure 18.

It can be seen from the figure that if the transverse micro-texture arrangement is adopted, it is possible to reduce the shear force only if it is arranged in a very narrow area at the outlet.

**Figure 18.** Influence of transverse texture position on dimensionless shear force on moving wall.
