Effect of snake-biomimetic surface texture on finger sealing performance under hydrodynamic lubrication

As shown in figure 1(a) finger seal consists of a front plate, a front spacer, 3 ∼ 5 layers of finger laminates, and a back plate. As the key parts of finger seals, each layer of the finger laminate composes of a sealing annulus and a series of finger curved beams starting from the inner circle and ending with finger feet. Moreover, each layer of finger laminate is interlaced and stacked together to eliminate the leakage channel along the gap of the finger beam. In general, it is a transition fit between the finger feet and the rotor, while high-speed rotation of the rotor, the compliant finger beam produces a hysteretic effect, resulting in the formation of a thin film between the finger feet and the rotor. Therefore, the fluid film cannot only achieve the sealing function but also avoid direct contact between the finger feet and the rotor, which can reduce the wear and improve the service life of the finger seal. In this paper, a series of surface textures are arranged on the rotor to improve the friction performance of the finger seal pair.

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As discussed in section 1, several bio-inspired surface textures have been studied to improve lubrication of friction pairs and wear resistance, as well as reduce viscosity and resistance. By considering the complex and harsh living environment, snakes have been considered as ideal biomimetic research objects for surface texture and friction properties [25, 26]. Moreover, the effective movement of snakes across different terrains is attributed to their characteristic scales [27, 28]. As snakes crawl, these scales can provide low friction with the ground and propel themselves forward in an 'S' shape. Furthermore, after million years of evolution, snakes' scales cannot only improve their mobility on the land (to control friction when crawling) and in the water (to reduce hydrodynamic resistance when swimming), but even in the air (to reduce aerodynamic resistance when flying) [29]. Therefore, as inspired by these tremendous features, snake scale shapes are adopted in the surface texture design of finger seals. In this paper, four typical types of snake scale shapes, i.e. Diamond, Hexagon, Ellipse, and Triangle, are selected as the texture shapes of the finger seal surface, as illustrated in figure 2.

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2.2.1. Modelling assumption

To study the influence of surface texture under hydrodynamic lubrication, a mathematical model is elaborated based on the Reynolds equation, which is well-known for the high consistency between theoretical analysis and experimental tests [30–33]. To establish the mathematical model, the following assumptions are made first:

• The effects of volume force (e.g. gravity) and inertial force (e.g. fluid acceleration) are ignored.
• No fluid sliding exists on the friction interface of the finger seal.
• Pressure change along the direction of lubrication film thickness is ignored.
• Between the frictional pair surface, the fluid is a Newtonian fluid and the flow is laminar, whilst the lubricating film is free of eddy current and turbulence.
• The influence of the deformation in the finger seal and rotor is ignored.
• The alignment between the finger seal and the rotor is ideal, which occurs with no disturbance of the sealing system.
• The surface between the finger feet and the rotor is smooth with no surface roughness.
• The variations of temperature difference and viscosity in the flow field are ignored.

Furthermore, considering that the seal clearance and texture depth are in micron-scale, while the circumference diameter of the rotor and the finger feet are in a millimeter-scale, it belongs to the cross-scale research scope. For the convenience of the research, the rotor and the finger feet are taken straight, and the rotating motion between them is replaced by a plane motion.

Based on the aforementioned assumption and description, the sketch of the proposed textured finger seal is proposed as shown in figure 3 and parameters are listed in table A1. It is assumed that an even load, p, is applied to the high-pressure finger laminate at the inlet boundary. The surface texture is constructed on the circumferential surface of the rotor, and the length and width of the rotor are x_b and y_b respectively. The sealing clearance between the rotor and the finger feet is h_f , the texture depth is h_w , and the surface texture density is S_p = S_w /S, where, S_w is the textured area, and S is the circumferential surface area of the rotor in the sealing section.

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2.2.2. Governing equation

Under the steady-state condition, the pressure governing equation of the hydrodynamic lubrication with surface texture can be written as:

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial x}\left(\displaystyle \frac{{h}^{3}}{\eta }\displaystyle \frac{\partial p}{\partial x}\right)+\displaystyle \frac{\partial }{\partial y}\left(\displaystyle \frac{{h}^{3}}{\eta }\displaystyle \frac{\partial p}{\partial y}\right)=6\displaystyle \frac{\partial }{\partial x}\left(Uh\right)\end{eqnarray} \tag{ 1 }$$

where x and y represent coordinates in a global Cartesian coordinate system, p is the fluid film pressure, h is the local film thickness at a specific point, η is the dynamic viscosity of the fluid, and U is the sum of the two interface velocities. Due to the surface texture, the thickness of the fluid film between the sealing pairs changes regularly and can be presented as follows:

$$\begin{eqnarray}h\left(x,y\right)=\left\{\begin{array}{ll}{h}_{f}+{h}_{w} & \left(x,y\in {{\rm{\Omega }}}_{w}\right)\\ {h}_{f} & \left(x,y\notin {{\rm{\Omega }}}_{w}\right)\end{array}\right.\end{eqnarray} \tag{ 2 }$$

where, Ω_w is the surface texture region.

2.2.3. Boundary conditions

• (1)  
  Forced boundary conditionsAs shown in figure 3, the left side of the rotor at x = 0 is defined as the pressure inlet, and its pressure is p_I (medium pressure difference), while the right side of the rotor at x = x_b is defined as the pressure outlet, and its pressure is p_O (set as 0 MPa). Therefore, the boundary conditions are as follows:

  $$\begin{eqnarray}&&\left\{\begin{array}{l}p\left(x=0,y\right)={p}_{I}\\ p\left(x={x}_{b},y\right)={p}_{O}\end{array}\right.\end{eqnarray} \tag{ 3 }$$
• (2)  
  Periodic boundary conditionsConsidering the high circumferential symmetry of the finger seal, a straightening process is carried out on the section of the rotor circumferential surface. As shown in figure 3, the simplified geometric model is symmetrical about the X-axis along the Y direction, namely, the pressures at upper and lower boundaries in the calculation domain are equal and the pressure gradient along the Y direction is equal. Hence, the periodic boundary conditions are obtained as follows:

  $$\begin{eqnarray}&&\left\{\begin{array}{l}p\left(x,y=\tfrac{1}{2}{y}_{b}\right)=p\left(x,y=-\tfrac{1}{2}{y}_{b}\right)\\ \displaystyle \frac{\partial p}{\partial y}\left(x,y=\tfrac{1}{2}{y}_{b}\right)=\displaystyle \frac{\partial p}{\partial y}\left(x,y=-\tfrac{1}{2}{y}_{b}\right)=0\end{array}\right.\end{eqnarray} \tag{ 4 }$$
• (3)  
  Cavitation boundary conditions

Since there is a convergent gap in the sealing interface, through the relative motion between the interfaces, the positive pressure of the fluid is generated in the convergent gap, while the negative one occurs in the divergent gap, therefore, a cavitation phenomenon will arise when the fluid pressure in the divergent region is lower than the cavitation pressure. For this, the cavitation phenomenon should be considered when studying the hydrodynamic effect of textured finger seal [34]. In this paper, the Reynolds cavitation boundary condition is adopted as follows:

$$\begin{eqnarray}&&{p}_{c}={p}_{a},\displaystyle \frac{\partial p}{\partial x}=\displaystyle \frac{\partial p}{\partial y}=0\,\left(x,y\right)\in {{\rm{\Omega }}}_{c}\end{eqnarray} \tag{ 5 }$$

where p_c is the cavitation pressure, p_a is the environmental pressure and ${{\rm{\Omega }}}_{c}$ is the cavitation region.

2.2.4. Dimensionless equation

To reduce the number of independent and dependent variables and improve the generality and stability of the calculation process, the dimensionless method is employed to solve the equation, where:

$$\begin{eqnarray}&&X=\displaystyle \frac{x}{{x}_{b}},Y=\displaystyle \frac{y}{{x}_{b}},H=\displaystyle \frac{h}{{h}_{f}},P=\displaystyle \frac{p}{{p}_{I}}\end{eqnarray} \tag{ 6 }$$

By substituting equation (6) into equation (1), the dimensionless form of Reynolds equation for finger seal with the continuous liquid flow is obtained as:

$$\begin{eqnarray}&&\displaystyle \frac{\partial }{\partial X}\left(\displaystyle \frac{{H}^{3}\partial P}{\partial X}\right)+\displaystyle \frac{\partial }{\partial Y}\left(\displaystyle \frac{{H}^{3}\partial P}{\partial Y}\right)={\rm{\Lambda }}\displaystyle \frac{\partial H}{\partial X}\end{eqnarray} \tag{ 7 }$$

where ${\rm{\Lambda }}=\tfrac{6U\eta {x}_{b}}{{h}_{f}^{2}{p}_{I}}$ is the operating parameter of finger seal.

Similarly, by substituting equation (6) into equations (2), and (6), the dimensionless form of the film thickness and boundary equations can be obtained as follows:

$$\begin{eqnarray}H\left(X,Y\right)=\left\{\begin{array}{ll}1+\displaystyle \frac{{h}_{w}}{{h}_{f}} & \left(X,Y\in {{\rm{\Omega }}}_{w}\right)\\ 1 & \left(X,Y\notin {{\rm{\Omega }}}_{w}\right)\end{array}\right.\end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray}&&\left\{\begin{array}{l}\begin{array}{l}P\left(X=0,Y\right)=1\\ P\left(X=1,Y\right)=\displaystyle \frac{{p}_{O}}{{p}_{I}}\end{array}\\ \begin{array}{l}P\left(X,Y=\displaystyle \frac{2{y}_{b}}{{x}_{b}}\right)=P\left(X,Y=-\displaystyle \frac{2{y}_{b}}{{x}_{b}}\right)\\ \displaystyle \frac{\partial P}{\partial Y}\left(X,Y=\displaystyle \frac{2{y}_{b}}{{x}_{b}}\right)=\displaystyle \frac{\partial P}{\partial Y}\left(X,Y=-\displaystyle \frac{2{y}_{b}}{{x}_{b}}\right)=0\\ {P}_{c}=P{}_{a},\displaystyle \frac{\partial P}{\partial X}=\displaystyle \frac{\partial P}{\partial Y}=0\,\left(X,Y\right)\in {{\rm{\Omega }}}_{c}\end{array}\end{array}\right.\end{eqnarray} \tag{ 9 }$$

2.2.5. Sealing performance parameters

Once the dimensionless pressure (P) is calculated and transformed into the pressure of the fluid film (p), the bearing capacity, friction force, and friction coefficient of the fluid film can be then calculated.

• (1)  
  Bearing force and average bearing pressure of the fluid filmThe bearing force W of the fluid film can be obtained by integrating p(x, y) in the whole fluid domain as:

  $$\begin{eqnarray}&&W=\displaystyle \int \displaystyle \int p\left(x,y\right)dxdy\end{eqnarray} \tag{ 10 }$$

  Then the average bearing pressure of the fluid film $\overline{p}$ can be calculated based on the bearing force W and the fluid domain area S as:

  $$\begin{eqnarray}&&\overline{p}=\displaystyle \frac{W}{S},\,S={x}_{b}\cdot {y}_{b}\end{eqnarray} \tag{ 11 }$$
• (2)  
  Friction force

By defining the fluid velocity at any point in the fluid film along the X direction as u and substituting into Newton's viscosity law, the shear stress of the fluid can be obtained as:

$$\begin{eqnarray}&&\tau =\eta \displaystyle \frac{\partial u}{\partial z}=\displaystyle \frac{1}{2}\displaystyle \frac{\partial p}{\partial x}\left(2z-h\right)+\left({U}_{h}-{U}_{0}\right)\displaystyle \frac{\eta }{h}\end{eqnarray} \tag{ 12 }$$

where U₀ and U_h are the surface velocities of the rotor and finger seal, respectively.

Therefore, when the finger seal is fixed and the rotor rotates, the shear forces at the upper and lower surfaces in the finger sealing area are as:

$$\begin{eqnarray}&&{\tau }_{h}=\displaystyle \frac{h}{2}\displaystyle \frac{\partial p}{\partial x}-{U}_{0}\displaystyle \frac{\eta }{h},\,{\tau }_{0}=-\displaystyle \frac{h}{2}\displaystyle \frac{\partial p}{\partial x}-{U}_{0}\displaystyle \frac{\eta }{h}\end{eqnarray} \tag{ 13 }$$

The upper and lower surface friction force can be then obtained as:

$$\begin{eqnarray}&&\left\{\begin{array}{l}{F}_{h}=\displaystyle {\int }_{0}^{{x}_{b}}\displaystyle {\int }_{0}^{{y}_{b}}\left(\displaystyle \frac{h}{2}\displaystyle \frac{\partial p}{\partial x}-{U}_{0}\displaystyle \frac{\eta }{h}\right)dxdy\\ {F}_{0}=\displaystyle {\int }_{0}^{{x}_{b}}\displaystyle {\int }_{0}^{{y}_{b}}\left(-\displaystyle \frac{h}{2}\displaystyle \frac{\partial p}{\partial x}-{U}_{0}\displaystyle \frac{\eta }{h}\right)dxdy\end{array}\right.\end{eqnarray} \tag{ 14 }$$

(3) Friction coefficient

After calculating the bearing force and friction force of the fluid film, the friction coefficient is obtained as:

$$\begin{eqnarray}&&\mu =\displaystyle \frac{F}{W}\end{eqnarray} \tag{ 15 }$$

In this paper, the finite difference method is adopted to solve the Reynolds equation. The successive overrelaxation method (SOR) [35] is employed to solve the linear equations. In the iterative process, it is calculated according to the following formula:

$$\begin{eqnarray}&&{P}^{k+1}\left(i,j\right)={P}^{k}\left(i,j\right)+\alpha \left({P}^{k+1}\left(i,j\right)-{P}^{k}\left(i,j\right)\right)\end{eqnarray} \tag{ 16 }$$

where, k is the iteration number and α is the relaxation factor.

The convergence accuracy is set as follows:

$$\begin{eqnarray}&&\displaystyle \frac{\displaystyle {\sum }_{i=1}^{m}\displaystyle {\sum }_{j=1}^{n}\left|{P}^{k+1}\left(i,j\right)-{P}^{k}\left(i,j\right)\right|}{\displaystyle {\sum }_{i=1}^{m}\displaystyle {\sum }_{j=1}^{n}\left|{P}^{k+1}\left(i,j\right)\right|}\leqslant Er{r}_{p}\end{eqnarray} \tag{ 17 }$$

where, Err_p is the error limit.

To perform the aforementioned numerical analysis, an iterative computation process is proposed as illustrated in figure 4 and carried out in the MATLAB^®. As the setting of grid density, relaxation factor and error limit directly affect the accuracy and efficiency of calculation, the trial calculation is carried out first taking the average dimensionless pressure and iteration steps as the analysis objectives. According to the calculation results, the accuracy and efficiency of the calculation can be satisfied when the relaxation factor, grid density and the error limit are set as 1.9, 200 × 200, and 10⁻⁵ respectively.

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In the practical study of surface texture in the sealing area, various microgroove or micropore textures are often configured in different amounts or patterns. While in the past theoretical research, to reduce the complexity, most researchers only adopt a single micro-texture element to establish geometric models, which ignores the actual geometry, boundary conditions, and the interaction between textures. In this section, to study the synergistic effect between surface textures, five different regions of texture elements along X and Y directions are considered as the calculation domain, i.e. 5 × 5, 3 × 5, 1 × 5, 1 × 3, and 1 × 1, as shown in figure 5.

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Based on the aforementioned analysis in section 2, the dimensionless pressure distributions in the five calculation domains are presented in figures 6(a)–(e), respectively. The mutual coupling effect exists between texture elements as well as the obvious hydrodynamic effects and cavitation phenomena in each calculation domain. Moreover, there is a dimensionless pressure peak in each surface textured micro-unit, which is caused by the convergence gap formed between the texture edge and the seal face with the motion direction of the rotor. Furthermore, due to the interaction between textures and the influence of boundary conditions, the film pressure distribution on each textured micro-unit is different.

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Figure 6(f) shows the comparison diagram of the maximum dimensionless pressure (P_max) and average dimensionless pressure (P_av ) in these five calculation domains, where P_max and P_av values are respectively plotted in charts and dots. Both P_max and P_av increase along with the growth of the number of textured microelements. While changing from 1 × 5 to 3 × 5, there is a significant increase both in the P_max and P_av values (the change rates are 14.07% and 46.4%, respectively). It is caused by the expansion of the texture matrix from a single dimension (in the X direction) to two dimensions (in the X and Y directions), which results in more intense coupling and synergy between textures.

Based on the above analysis, a calculation domain with 0.5 mm × 0.5 mm is adopted to study the dynamic pressure distribution of the finger seal with different types of texture, and 5 × 5 micro-textures are evenly distributed in the domain.

To analyze the influence of four snakeskin-inspired texture types on the hydrodynamic pressure, a series of texture arrangements are considered as listed in table 2. Moreover, the orientations of different textures are divided into transverse (perpendicular to the direction of rotor motion) and longitudinal (parallel to the direction of rotor motion). Furthermore, the area of a single micro-texture is set as 1472.58 μm².

Table 2. Geometrical parameters of surface texture.

Texture type	Arrangement form	Geometric parameter (μm)	Area (μm2)
Ellipse	E1	Long axis = 50;
	E2	Short axis = 37.5
Diamond	D1	Long diagonal = 62.66;
	D2	Short diagonal = 47	1472.58
Hexagon	H1	Side length = 23.8
	H2
Triangle	T1
	T2	Side length = 58.32
	T3

Figure 7 shows the dimensionless pressure (P) and friction coefficient (μ) of four different texture types with different orientations under the conditions of: rotor speed of 20000 r min⁻¹, inlet/outlet pressure difference of 0.3 MPa, texture area ratio of 14.73%, seal clearance and texture depth of 5 μm. Different textures and orientations play an important role in the dynamic pressure ability to seal fluid and the detailed analyses are as follows:

• (1)  
  The P_max and P_av of the longitudinal Ellipse (E2), Diamond (D2), and inverted Triangle (T2) textures are larger than those of transverse counterparts. This is because most of the profile of the longitudinal textures is consistent with the rotation direction of the rotor, and the textured wall has less obstruction to the overflow of the fluid medium, which has a good replenishment lubrication ability; Meanwhile, the bearing capacity is improved as results from the small flow resistance of the lubricating medium, the fast flow velocity, and the increased hydrodynamic pressure. However, in the transverse textures, there are more walls approximately perpendicular to the rotor rotation direction, which aggravates the obstruction to the flow of the fluid medium, weakens the dynamic pressure of the fluid, and reduces the bearing capacity.
• (2)  
  The trends of the P_max and P_av of Hexagon and Triangle texture are different. For example, the P_av of T2 is the largest (2.458), followed by T1 (2.378) and T3 (2.015). The P_max of T3 and T2 are similar, which are 5.794 and 5.751 respectively, while the P_max of T1 is 6.243. The reason for this phenomenon is that the cross-section size of T1 and T3 gradually decreases in the direction of rotor motion. When the fluid flows out of the textured wall, the wedge effect increases, which hinders the flow of the fluid medium. Therefore, the P_max of the fluid increases, while the overall dynamic pressure decreases.
• (3)  
  The aspect ratios of the Ellipse, Diamond, Hexagon, and Triangle textures are 1.333, 1.333, 1.155, and 1.155, respectively. The P_av of longitudinal texture is 1.143, 1.094, 0.977, and 1.033 times that of transverse texture. The effect of texture orientation on hydrodynamic pressure is closely related to the aspect ratio of texture. The P_max analysis is identical. Furthermore, among these four textures, it is found that the Ellipse and Diamond textures have the larger P and the smaller μ as compared with the Hexagon and the Triangle ones, which shows that the Ellipse and Diamond textures have the better anti-friction and anti-wear ability.

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Based on the above analysis, when the texture profile is consistent with the rotor rotation direction, it is conducive to the formation of better hydrodynamic pressure and the improvement of the sealing performance. Therefore, the longitudinal textures (i.e. E2, D2, H2 and T2) are chosen for further analysis in the following sections.

Figure 8 shows the P_av and μ of the four textured finger seals with various texture depths at two different rotor speeds (i.e. 20000 r min⁻¹ and 15000 r min⁻¹) under conditions of: inlet/outlet pressure difference of 0.3 MPa, seal clearance of 5 μm, and the texture area ratio of 14.73%. In general, the P_av is lower and the μ is higher without texture (depth = 0) as compared with the ones with texture (depth > 0). Then, as the texture deepens, the influence of four types of textures follows a similar tendency, in which the P_av increases sharply at first, then decreases and gradually flattens, while the μ is opposite. It's worth noting that when the texture depth (h_w ) is 5 μm, both performance indexes of four textured finger seals reach the extreme values, which indicates that the four textured finger seals can have good anti-friction and anti-wear properties in this case. Moreover, among the four textures, the Diamond texture always results in higher P_av and lower μ, which shows its best friction reduction and wear resistance performance under this working condition; While the Hexagon texture performs in an opposite way when the h_w is more than 10 μm.

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Besides, to obtain good friction reduction and anti-wear performance of textured finger seals, the h_w should be as small as possible (h_w < 25 μm). Taking the Hexagon texture with the rotation speed of 15000r min⁻¹ as an example, when the h_w is 25 μm, the P_av and the μ are 125.19% and 79.66% of those without texture, respectively; While the h_w is 30 μm, the P_av and μ are 98.1% and 101.68.73% of those without texture, respectively. In this case, the texture loses the effect of dynamic pressure, friction reduction, and anti-wear.

To further analyze the influence of h_w on finger sealing performance, the factor of sealing clearance (h_f ) is taken into account. In figure 9, the effect of the h_f of 1 ∼ 5 μm is presented under the condition of Ellipse h_w varying from 0 to 25 μm. It shows that: (1) the smaller the h_f is, the larger the P_av , and as the h_f increases, the variation trend of the P_av and the μ gradually becomes gentle; (2) the P_av value reaches the maximum (while the μ is minimum) when the h_w is the same as the h_f , which also reflects the effects at extreme values when h_w equals to h_f at 5 μm in figure 8; (3) when the h_w is in the range of 1 to 5 μm, the smaller the h_f is, the smaller the minimum value of μ is (e.g. at 20,000 r min⁻¹, the minimum value of μ−1 is 0.088, and the minimum value of μ−5 is 0.429, which is 4.875 times of that of μ−1). Therefore, the maximum P_av and the minimum μ can be obtained by reducing the h_f as well as matching the condition of h_w = h_f , which results in both the optimal sealing performance and anti-wear performance.

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To study the effect of texture density (S_p ) on the friction and sealing performance of the finger seal, the four textures are arranged interlaced to increase the S_p as much as possible, as shown in figure 10 (the S_p is 60%, and the corresponding texture geometric parameters are shown in table 3).

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Figure 11 shows the P_av and μ of the four textured finger seals with different textural surface densities under the identical conditions of: the rotor speed of 20,000 r min⁻¹, inlet/outlet pressure difference of 0.3 MPa and 0.5 MPa, seal clearance and texture depth of 3 μm. With the increase of S_p , the P_av increases and gradually flattens, while the μ is the opposite. The reason for this phenomenon is that the surface texture provides a regular convergence gap between the rotor and the finger feet. When the fluid film enters the texture region, it produces positive pressure at the convergence gap, and the pressure of the fluid film decreases at the divergence gap, even negative pressure. Thus, the asymmetric pressure distribution in each micro-texture region is generated, which results in the fluid film with certain bearing capacity. With the increase of S_p , the convergence gap area becomes larger, so the hydrodynamic effect is enhanced and the P_av is increased. When the S_p reaches a certain stage, the increase of S_p has no obvious effect on the rise of hydrodynamic pressure, so the P_av increases slowly, and the μ decreases slowly. Considering the manufacturing economy, the suitable S_p is around 20% –40%.

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Besides, the μ of Triangle texture is the largest and the P_av is the lowest, which indicates that the friction reduction and anti-wear performance of Triangle texture are the worst. In the appropriate range of S_p , Ellipse texture has the smallest μ and the largest P_av , which shows the best anti-wear performance, followed by Diamond and Hexagon textures.

Furthermore, comparing the four textures, the trend of the P_av -D and μ-D are different from the other three textures. When the S_p is greater than 40%, the P_av -D decreases and μ-D increases with the increase of S_p , while the other three textures are opposite. The reason for this phenomenon is that, when the S_p reaches a certain degree, the Dimond texture is more likely to form a connection trend in the direction of rotor motion (e.g. Y-axis in figure 10), resulting in more intense of fluid pressure relief, which leads to a decrease of the P_av -D and an increase of the corresponding μ-D.