## 2 Thermohydrodynamic Lubrication Model
The expression for the friction force caused by the contact of micro-asperities is shown in Equation (6):  
\[F_{A}=\tau_{0}A_{c}+\alpha W_{A} \tag{6}\]  
where \(\tau_{0}\) is the shear stress constant, \(\alpha\) is the boundary friction coefficient. The temperature of the oil film is influenced by multiple factors, with the surrounding environmental temperature directly affecting the oil film and leading to temperature changes. The heat generated during equipment operation is transmitted to the oil film, causing temperature fluctuations. When calculating the viscosity and density of the oil film, it is necessary to consider changes in the oil film temperature. Additionally, since the lubrication state of the piston ring-cylinder liner assembly is often mixed lubrication, the contribution of micro-asperity contact to heat generation should also be included in the energy equation, resulting in a modified formula shown in Equation (7):  
\[\hat{Q}_{asp}=|U|\frac{dF_{A}}{dV}=|U|\frac{P_{asp}k_{asp}dA}{dV}=\frac{P_{ asp}k_{asp}|U|}{H} \tag{7}\]  
where \(F_{A}\) is Friction force caused by asperity contacts, \(k_{asp}\) is the Coefficient of friction for asperity contact. U represents the sliding velocity of asperity contact. This parameter is used to calculate the friction and heat generated during asperity contact, which helps to evaluate its effect on the lubricant film temperature. Therefore, after considering the impact of micro-asperity contact on the oil film temperature, the energy equation is expressed as shown in Equation (8):  
\[ \begin{split}& k_{p}\frac{\partial^{2}T}{\partial z^{2}}=-\mu \left[\left(\frac{\partial u}{\partial z}\right)^{2}+\left(\frac{\partial v}{ \partial z}\right)^{2}\right]-\beta_{T}T\left(u\frac{\partial p}{\partial x}+v \frac{\partial p}{\partial y}\right)+\\ &\rho c_{p}\left(u\frac{\partial T}{\partial x}+v\frac{\partial T }{\partial y}+\frac{\partial T}{\partial z}\right)-\frac{P_{asp}k_{asp}|u|}{H} \end{split}  \tag{8}\]  
The piston ring directly contacts the oil film, transferring the heat from the piston and the high-temperature combustion gases within the cylinder to the oil film, which then conveys the heat to the cylinder liner, thus facilitating the heat transfer between the piston and the cylinder liner. The temperature control equation for the piston ring is determined by [PERSON]'s equation, as shown in Equation (9):  
\[k_{r}\left(\frac{\partial^{2}T}{\partial x^{2}}+\frac{\partial^{2}T}{\partial y^ {2}}+\frac{\partial^{2}T}{\partial z^{2}}\right)=\rho_{r}c_{r}\frac{\partial T }{\partial x} \tag{9}\]  
where \(\rho_{r}\) is the density of the piston ring, \(c_{r}\) is the specific heat capacity of the piston ring, \(k_{r}\) is the thermal conductivity of the piston ring. The oil film thickness formed between the friction pairs in the piston ring-cylinder liner assembly during operation is mainly influenced by the following factors: First, the loss of original roundness of the cylinder liner during operation increases the oil film thickness, second, the profile of the piston ring can increase the oil film thickness, third, the surface texture created during manufacturing also affects the oil film thickness. (1) Cylinder Liner Out-of-Roundness  
As shown in Figure 2, considering the elastic factors of the piston ring, the expression for the oil film thickness variation at time t due to the deformation of the inner wall surface of the cylinder liner is given by Equation (10):  
\[C(\omega,t)=\left\{ \begin{aligned} \Delta_{c}(\omega,t)-u_{N}(\omega,t), \Delta_{c}(\omega,t)&>u_{N}(\omega,t)\\ 0,&\Delta_{c}(\omega,t)&<u_{N}(\omega,t) \end{aligned} \right. \tag{10}\]  
(2) Piston Ring Profile  
To enhance the oil scraping function of the first compression ring on the cylinder liner surface, the outer surface of the first piston ring is designed with a specific profile rather than being flat. This profile improves the oil-scraping effect of the piston ring, but it also affects the oil film thickness formed between the cylinder liner and the piston ring.