# Structural Design and Optimization of Herringbone Grooved Journal Bearings Considering Turbulent
## 2 Numerical Model and Digital Simulation
### Establishment of Reynolds Equation
#### 2.2.2 Static Reynolds Equation II  
When dividing the mesh, the mesh boundary should coincide with the bearing ridge and groove boundary, so that the calculation accuracy is higher, and so the mesh shape is a parallelogram. Figure 3 shows that there are positions where the film thickness has a step change, which is difficult to calculate. At the same time, there is a cross partial derivative term in the equation where the film thickness is continuous, so the previous five-point difference method is no longer applicable. In this paper, the local integration method is used to study the continuous area of the oil film thickness and the step change of the oil film thickness to explain the principle of solving the equation.  
Figure 2: Schematic diagram of coordinate conversion.  
(1) Continuous area of the oil film thickness:  
For areas A and D, the oil film thickness is continuous. The integral area of the node \((i,j)\) where the film thickness is continuous is the small parallelogram region in the middle of the Figure 4.  
Integrate both sides of the static Reynolds equation at the same time [5]:  
\[\iint\limits_{D_{i,j}}\left[\frac{\partial}{\partial\varphi}\left(\frac{H^{3}}{ k_{x}}\frac{\partial p}{\partial\varphi}\right)+\left(\frac{d}{b}\right)^{2} \frac{\partial}{\partial\lambda}\left(\frac{H^{3}}{k_{x}}\frac{\partial p}{ \partial\lambda}\right)\right]d\varphi d\lambda=\frac{1}{4}\iint\limits_{D_{ i,j}}\left(\frac{\partial H}{\partial\varphi}\right)d\varphi d\lambda \tag{14}\]  
Use [PERSON]'s formula to convert to line integral form:  
\[\int\limits_{L_{i,j}}\frac{H^{3}}{k_{x}}\frac{\partial p}{\partial\varphi}d \lambda-\left(\frac{d}{b}\right)^{2}\int\limits_{L_{i,j}}\frac{H^{3}}{k_{x}} \frac{\partial p}{\partial\lambda}d\varphi=\frac{1}{4}\int\limits_{L_{i,j}}Hd\lambda \tag{15}\]  
Substitute the coordinate transformation Expression (11) and the pressure deviation Expression (12) into the above formula, and the five-point difference formula of the oil film continuum area under the oblique coordinates can be obtained as follows:  
\[P_{i,j}=\frac{A_{i,j}P_{i+1,j}+B_{i,j}P_{i-1,j}+C_{i,j}P_{i,j+1}+D_{i,j}P_{i,j -1}-F_{i,j}}{E_{i,j}} \tag{16}\]  
In the formula:  
\[A_{i,j}=\frac{\Delta x}{\Delta\eta}\sin\alpha\frac{H^{3}_{i+0.5,j}}{k_{x}}- \left(\frac{d}{b}\right)^{2}\left(\frac{-H^{3}_{i,j-0.5}+H^{3}_{i,j+0.5}}{2k_{ z}\tan\alpha}-\frac{\Delta\xi}{\Delta\eta}\frac{\cos\alpha}{\tan\alpha}\frac{H^{3} _{i+0.5,j}}{k_{z}}\right)\]  
\[B_{i,j}=\frac{\Delta x}{\Delta\eta}\sin\alpha\frac{H^{3}_{i-0.5,j}}{k_{x}}- \left(\frac{d}{b}\right)^{2}\left(\frac{H^{3}_{i,j-0.5}-H^{3}_{i,j+0.5}}{2k_{ z}\tan\alpha}-\frac{\Delta\xi}{\Delta\eta}\frac{\cos\alpha}{\tan\alpha} \frac{H^{3}_{i-0.5,j}}{k_{z}}\right)\]  
Figure 4: Continuous area of the oil film thickness.  
Figure 3: Schematic diagram of meshing.  
\[C_{i,j}=-\left(\frac{d}{b}\right)^{2}\left(-\frac{\Delta\eta}{\Delta\xi}\frac{1}{ \sin\alpha}\frac{H_{i,j+0.5}^{3}}{k_{z}}-\cos\alpha\frac{-H_{i+0.5,j}^{3}+H_{i- 0.5,j}^{3}}{2k_{z}\sin\alpha}\right)\]  
\[D_{i,j}=-\left(\frac{d}{b}\right)^{2}\left(-\frac{\Delta\eta}{\Delta\xi}\frac{1} {\sin\alpha}\frac{H_{i,j-0.5}^{3}}{k_{z}}-\cos\alpha\frac{H_{i+0.5,j}^{3}-H_{i -0.5,j}^{3}}{2k_{z}\sin\alpha}\right)\]  
\[E_{i,j}=A_{i,j}+B_{i,j}+C_{i,j}+D_{i,j}\]  
\[F_{i,j}=\frac{1}{4}\sin\alpha\Delta\xi\left(H_{i+0.5,j}-H_{i-0.5,j}\right)\]  
(2) Noncontinuous area of the oil film thickness:  
For the noncontinuous area of the oil film thickness, such as area B, the node falls on the boundary between the ridge and the groove, and the film thickness has a step change, as shown in Figure 5. Integrate the areas on both sides of the node \((i,j)\) separately, and use the same processing method as the continuous area of the oil film thickness. The five-point difference formula of the oil film discontinuous area under the oblique coordinate is as follows:  
\[P_{i,j}=\frac{A_{i,j}^{\prime}P_{i+1,j