# Comprehensive Investigations on Fluid Flow and Cavitation Characteristics in Rotating Disk System
## 2 Numerical Analysis
### Reynolds Equation Dimensionlessification  
As shown in Figure 5, the basic idea of the finite difference algorithm is to replace the continuous fixed solution region with a grid composed of a finite number of discrete points, which are used as nodes of the grid. For the oil film discussed above, the \(r\) direction is divided into m cells and the \(\theta\) direction is uniformly divided into \(n\) cells, then the radial and circumferential steps of the oil film are \(\Delta r=0.005/\)m and \(\Delta\theta=12/n\), respectively.  
Differentiate the dimensionless Reynolds equation in the discrete solution domain.  
\[\begin{array}{l}H^{3}\frac{\partial P}{\partial R}+R\frac{\partial}{ \partial R}\left(H^{3}\frac{\partial P}{\partial R}\right)+\frac{1}{4\pi^{2}R }\frac{\partial}{\partial\Phi}\left(H^{3}\frac{\partial P}{\partial\Phi} \right)\\ =K_{1}\bigg{[}R\frac{\partial(H)}{\partial\Phi}+K_{2}\bigg{(}2 RH^{3}+R^{2}\frac {\partial(H^{3})}{\partial R}\bigg{)}\bigg{]}\end{array} \tag{12}\]  
where the first-order partial derivatives and second-order partial derivatives are denoted as:  
\[\left\{\begin{array}{l}\frac{\partial P}{\partial R}=\frac{P_{(i+1)}-P_{(i- 1)}}{2\Delta R}\\ \frac{\partial P}{\partial R}=\frac{P_{(i+1)}-P_{(i-1)}}{2\Delta R}\end{array}\right. \tag{13}\]  
Figure 4: Film thickness under uniformly distributed hexahedron texture.  
Figure 5: Solution domain discretization of finite difference algorithm (The black circles are adjacent discrete points.).  
\[\left\{\begin{array}{c}\frac{\partial}{\partial R}\left(H^{3}\frac{\partial p}{ \partial R}\right)=\frac{H_{(i+1/2,i)}^{3}P_{(i+1,i)}+H_{(i-1/2,i)}^{3}P_{(i-1,i )}-\left(H_{(i+1/2,i)}^{3}+H_{(i-1/2,i)}^{3}\right)P_{(i,i)}}{\left(\Delta R \right)^{2}}\\ \frac{d}{\partial R}\left(H^{3}\frac{\partial p}{\partial R}\right)=\frac{H_{ (i+1/2)}^{3}P_{(i+1)}+H_{(i(j-1/2)}^{3}P_{(i-j-1)}-\left(H_{(i,j+1/2)}^{3}+H_{ (i,j-1/2)}^{3}\right)P_{(i,j)}}{\left(\Delta\Phi\right)^{2}}\end{array}\right. \tag{14}\]  
Substituting Equations (13) and (14) into Equation (12) can obtain the dimensionless Reynolds equation in the differential form:  
\[A_{(i,j)}P_{(i+1,j)}+B_{(i,j)}P_{(i-1,j)}+C_{(i,j)}P_{(i,j+1)}+D_{(i,j)}P_{(i, j-1)}-E_{(i,j)}P_{(i,j)}=F_{(i,j)} \tag{15}\]  
where \(A_{(i,j)}\sim F_{(i,j)}\) are the discrete point coefficients:  
\[A_{(i,j)}=\frac{H_{(i,j)}^{3}}{2\Delta R}+\frac{R_{(i,j)}H_{(i+1/2,j)}^{3}}{ \left(\Delta R\right)^{2}} \tag{16}\]  
\[B_{(i,j)}=-\frac{H_{(i,j)}^{3}}{2\Delta R}+\frac{R_{(i,j)}H_{(i-1/2,j)}^{3}}{ \left(\Delta R\right)^{2}} \tag{17}\]  
\[C_{(i,j)}=\frac{1}{4\pi^{2}R_{(i,j)}}\frac{H_{(i,j+1/2)}^{3}}{\left(\Delta \Phi\right)^{2}} \tag{18}\]  
\[D_{(i,j)}=\frac{1}{4\pi^{2}R_{(i,j)}}\frac{H_{(i,j-1/2)}^{3}}{\left(\Delta \Phi\right)^{2}} \tag{19}\]  
\[E_{(i,j)}=\frac{R_{(i,j)}\left(H_{(i+1/2,j)}^{3}+H_{(i-1/2,j)}^{3}\right)}{ \left(\Delta R\right)^{2}}+\frac{H_{(i,j+1/2)}^{3}+H_{(i,j-1/2)}^{3}}{4\pi^{2} R_{(i,j)}\left(\Delta\Phi\right)^{2}} \tag{20}\]  
\[F_{(i,j)}=K_{1}\left[R_{(i,j)}\frac{H_{(i,j+1)}-H_{(i,j+1)}}{2\Delta\Phi}+K_{ 2}\left[2R_{(i,j)}H_{(i,j)}^{3}+R_{(i,j)}^{2}\frac{H_{(i+1,j)}^{3}-H_{(i-1,j)} ^{3}}{2\Delta R}\right]\right] \tag{21}\]  
The pressure approximation solution \(P_{(i,j)}\) of the Reynolds equation can be expressed as:  
\[P_{(i,j)}^{k}=w\frac{A_{(i,j)}P_{(i+1,j)}+B_{(i,j)}P_{(i-1,j)}+C_{(i,j)}P_{(i, j+1)}+D_{(i,j)}P_{(i,j-1)}-F_{(i,j)}}{E_{(i,j)}}+(1-w)P_{(i,j)}^{k-1} \tag{22}\]  
The relevant parameters are solved to calculate the velocity and pressure in the oil film, and the position prediction of cavitation incipient and the corresponding calculation flowchart is shown in Figure 6.