## 2 Thermohydrodynamic Lubrication Model
The oil film thickness at a point (\(x\),\(y\)) on the outer surface of the piston ring can be expressed as shown in Equation (11):  
\[h_{r}(x,y)=\frac{4\delta x^{2}}{b^{2}} \tag{11}\]  
where \(\delta\) is the barrel-shaped height of the piston ring's outer surface, \(b\) is the axial width of the piston ring, \(x\) is the horizontal coordinate of the point under an absolute coordinate system. (3) Surface Texture  
By comprehensively considering the interaction between cylinder liner deformation and piston ring elasticity, the piston ring outer surface profile, and the cylinder liner surface texture, a lubrication model is established to accurately describe the oil film thickness between the compression ring and the cylinder liner. Thus, the expression for the variation  
Figure 2: Relationship between deformation of cylinder liner and piston ring and thickness of oil film. of oil film thickness at a point (\(x\),\(y\)) between the deformed cylinder liner and the compression ring is shown in Equation (12):  
\[h(x,y)=\left\{ \begin{matrix}h_{0}+C(\omega,t)+h_{r}(x,y)+h_{text}(x,y),x,y\in \Omega\\ h_{0}+C(\omega,t)+h_{r}(x,y),\hskip 28.452756 ptx,y\in\Omega\end{matrix} \right. \tag{12}\]  
This study mainly investigates the lubrication characteristics of the first compression ring, with its structural and material parameters listed in Table 1.  
A lubrication calculation model is established for the textured cylinder liner and the first compression ring, with specific surface microtopography parameters and piston ring-cylinder liner data shown in Table 2.  
To evaluate the friction loss and wear of the textured compression ring-cylinder liner system, the Friction Mean Effective Pressure (FMEP) and wear load are considered, with their expressions given by Equations (13) and (14):  
\[FMEP=\int\frac{f_{total}}{V_{d}}\mathrm{d}l_{s} \tag{13}\]  
\[W_{load}=\frac{1}{\theta_{0}}\int_{0}^{\theta_{0}}\int P_{asp}|U|\mathrm{d}\theta \tag{14}\]  
where \(W_{load}\) is the wear load, \(l_{s}\) is the piston stroke, \(P_{asp}\) is the micro-asperity contact pressure, \(V_{d}\) is the displacement volume of a single cylinder in the internal combustion engine, \(\theta_{0}\) is the crank angle corresponding to the mechanical cycle. Under boundary and mixed lubrication conditions, the lubricant film reaches the nanometer scale, and the micro-elastohydrodynamic effect significantly influences the reliability and performance of the friction surface. \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter Name** & **Value** \\ \hline Cylinder Diameter D/mm & 126 \\ Crank Radius R/mm & 65 \\ Connecting Rod Length l/mm & 200 \\ Rotation Speed n/ \(\left(\mathrm{r}\cdot\mathrm{min}^{-1}\right)\) & 2000 \\ Lubricant Grade & 15W-40 \\ Lubricant Density \(\rho/\left(\mathrm{kg}\cdot\mathrm{m}^{-3}\right)\) & 887 \\ Viscosity at Reference Temperature \(\mu_{0}/\mathrm{Pa}\cdot\mathrm{s}\) & 1.2 \(\times\) 10\({}^{-4}\) \\ Lubricant Specific Heat Capacity \(c_{p}/\left(\mathrm{J}\cdot\left(\mathrm{kg}\cdot\mathrm{K}\right)^{-1}\right)\) & 1880 \\ Lubricant Thermal Conductivity W/(m\(\cdot\)K) & 0.12 \\ Viscosity–Temperature Coefficient \(\alpha_{2}\) & 0.042 \\ Viscosity–Pressure Coefficient \(\alpha_{1}\) & 2.2 \(\times\) 10\({}^{-8}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Basic calculation parameters. \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter Name** & **Value** \\ \hline Texture Depth \(h_{p}/\mathrm{\SIUnitSymbolMicro m}\) & 5–10 \\ Spherical Texture Radius \(r_{p}/\mathrm{\SIUnitSymbolMicro m}\) & 25–40 \\ Texture Density \(s_{p}\) & 20–40\% \\ Cylinder Liner Surface Roughness \(\sigma_{1}/\mathrm{\SIUnitSymbolMicro m}\) & 0.4 \\ Piston Ring Surface Roughness \(\sigma_{r}/\mathrm{\SIUnitSymbolMicro m}\) & 0.4 \\ Piston Ring Barrel-Shaped Height \(\delta/\mathrm{\SIUnitSymbolMicro m}\) & 20 \\ Piston Ring Axial Width \(b/\mathrm{\SIUnitSymbolMicro m}\) & 1.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Textured surface structural parameters.