# Study on the Performance of Liquid Metal Lubricated V-Groove Bearing Considering Turbulence
## 3 Numerical Model and Simulation
### Establishment of Reynolds Equation for V-G B  
Considering that the Ng-Pan turbulent model based on the law of the wall is the most widely used turbulent model. Meanwhile, its turbulent factor is linear, which makes the calculation more simple and suitable for light load hydrodynamic sliding bearing. Therefore, by choosing this model, the dimensionless Reynolds equation is as follows:  
\[\frac{\partial}{\partial\varphi}\left(\frac{H^{3}}{k_{\rm x}}\frac{\partial P}{ \partial\varphi}\right)+\left(\frac{d}{b}\right)^{2}\frac{\partial}{\partial \lambda}\left(\frac{H^{3}}{k_{\rm z}}\frac{\partial P}{\partial\lambda}\right) =\frac{1}{4}\frac{\partial H}{\partial\varphi} \tag{2}\]  
where \(k\) is the turbulent factor and the value is  
\[\left\{\begin{array}{c}k_{\rm x}=12+0.0136(Re)^{0.9}\\ k_{\rm z}=12+0.0043(Re)^{0.98}\end{array}\right. \tag{3}\]  
It can be seen from Figure 2 that the V-groove's structure is special because its trend is inclined in a straight line. The rectangular grid under the Cartesian coordinate system (CCS) will increase the difficulty in dealing with the flow discontinuity (i.e., at the groove step). Therefore, transforming coordinate, the calculation, and analysis were carried out in the oblique coordinate system (OCS) consistent with the groove direction. Figure 3 is a diagram of the coordinate transformation.  
As shown in Figure 3, with the inclination angle of half of the included angle of the V-groove, the CCS was transformed into the OCS. The corresponding calculation relationship is  
\[\left\{\begin{array}{c}\varphi=\eta-\xi\cos\left(\frac{\alpha}{2}\right)\\ \lambda=\xi\sin\left(\frac{\alpha}{2}\right)\end{array}\right.\Leftrightarrow \left\{\begin{array}{c}\eta=\varphi+\frac{\lambda}{\tan\left(\frac{\alpha}{ 2}\right)}\\ \xi=\frac{\lambda}{\sin\left(\frac{\alpha}{2}\right)}\end{array}\right. \tag{4}\]  
Figure 3: Diagram of coordinate transformation.  
Figure 2: The structural diagram of V-g B.  
The expression of pressure partial derivative is obtained as followed:  
\[\left\{\begin{array}{c}\frac{\partial P}{\partial\varphi}=\frac{\partial P}{ \partial\eta}\frac{\partial\eta}{\partial\varphi}+\frac{\partial P}{\partial \varphi}\frac{\partial\varphi}{\partial\varphi}+\frac{\partial P}{\partial \varphi}\frac{\partial\varphi}{\partial\varphi}=\frac{\partial P}{\partial \eta}\\ \frac{\partial P}{\partial\varphi}=\frac{\partial P}{\partial\eta}\frac{ \partial\eta}{\partial\lambda}+\frac{\partial P}{\partial\varphi}\frac{ \partial\varphi}{\partial\lambda}=\frac{1}{\tan\frac{1}{2}\frac{\partial P }{\partial\eta}}+\frac{1}{\sin\frac{1}{2}\frac{\partial P}{\partial\xi}}\end{array}\right. \tag{5}\]  
Then, bring Equation (5) into Equation (2), and the Reynolds equation in OCS is as follows:  
\[\begin{array}{c}\frac{\partial}{\partial\eta}\left(\frac{H^{3}}{k_{x}} \frac{\partial P}{\partial\eta}\right)+\frac{d^{2}}{p^{2}\tan^{2}\left(\frac{ \alpha}{2}\right)}\frac{\partial}{\partial\eta}\left(\frac{H^{3}}{k_{x}}\frac {\partial P}{\partial\eta}\right)+\frac{d^{2}}{p^{2}\tan\left(\frac{\alpha}{2 }\right)\sin\left(\frac{\alpha}{2}\right)}\frac{\partial}{\partial\eta}\left( \frac{H^{3}}{k_{x}}\frac{\partial P}{\partial\varphi}\right)\\ +\frac{d^{2}}{p^{2}\tan\left(\frac{\alpha}{2}\right)\sin\left(\frac{\alpha}{2} \right)}\frac{d^{2}}{\partial\xi}\left(\frac{H^{3}}{k_{x}}\frac{\partial P}{ \partial\eta}\right)+\frac{d^{2}}{p^{2}\sin^{2}\left(\frac{\alpha}{2}\right)} \frac{\partial}{\partial\xi}\left(\frac{H^{3}}{k_{x}}\frac{\partial P}{\partial \xi}\right)=\frac{1}{4}\frac{\partial H}{\partial\eta}\end{array} \tag{6}\]  
At this time, the oil film thickness equation is:  
\[H=1+\varepsilon cos\varphi=1+\varepsilon cos\left(\eta-\xi cos\frac{\alpha}{2 }-\theta\right) \tag{7}\]  
groove:  
\[H=1+\varepsilon cos\varphi+\gamma_{g}=1+\varepsilon cos\left(\eta-\xi cos \frac{\alpha}{2}-\theta\right)+\gamma_{g} \tag{8}\]  
where \(\gamma_{g}=\frac{cg}{c}\) is the groove depth ratio; \(\varepsilon=\frac{\varepsilon}{c}\) is the eccentricity; \(cg\) is the groove depth; and \(c\) is the radius gap.