## 2 Lubrication Model and Algorithm of the Regional Flow Pattern
### Governing Equations  
1. The Reynolds equation  
As shown in Figure 2, the pad surface of the WTTB is fan-shaped, and the pad can be tilted around the fulcrum. \(R_{1}\) and \(R_{2}\) are the inner and outer radii of the pad, respectively. \(\theta_{0}\) is the pad sector angle. The Reynolds equation considering the flow pattern and ignoring the inertial force effect according to [18,19] is as follows:  
\[\frac{\partial}{\partial r}(G_{r}\frac{rh^{3}}{\mu}\frac{\partial p}{\partial r })+\frac{1}{r}\frac{\partial}{\partial\theta}(G_{\theta}\frac{h^{3}}{\mu} \frac{\partial p}{\partial\theta})=\frac{r\
u}{2}\frac{\partial h}{\partial\theta} \tag{2}\]  
where \(G_{r}\) and \(G_{\theta}\) are the flow pattern coefficients. In the laminar flow region, \(G_{r}=G_{\theta}=1/12\). In the turbulent region, the coefficients [7] are as follows:  
\[\left\{ \begin{array}{l}G_{\theta}=(12+0.0136R_{c}^{0.9})^{-1}\\ G_{r}=(12+0.0043R_{c}^{0.96})^{-1}\end{array} \right. \tag{3}\]  
Figure 1: Partitioning flow pattern of the WTTB. Figure 2: Structure schematic of the WTTB: (**a**) tilted state and (**b**) geometry. In the turbulent region, \(h\) is the average film thickness. The boundary condition is as follows:  
\[\left\{ \begin{array}{ll}p_{j}=0&j\in\Gamma_{1},\;\Gamma_{1}:&\text{Pad boundary}\\ p_{j}=0,&\frac{\partial p_{j}}{\partial\theta_{j}}=0\;j\in\Gamma_{2},\;\Gamma_{2 }:&\text{Water film rupture boundary}\\ p_{j\mathbb{L}}=p_{j\mathbb{T}}&j\in\Gamma_{3},\Gamma_{3}:&\text{Interface between laminar and turbulent flow}\\ \end{array} \right.\] 2. The energy equation  
The steady state two-dimensional energy equation considering the flow pattern according to [20] is as follows:  
\[\rho C_{p}\bigg{[}\bigg{(}\frac{\omega rh}{2}-G_{\theta}\frac{h^{3}}{12\mu} \frac{1}{r}\frac{\partial p}{\partial\theta}\bigg{)}\frac{1}{r}\frac{\partial t }{\partial\theta}-G_{r}\frac{h^{3}}{12\mu}\frac{\partial p}{\partial r}\frac{ \partial t}{\partial r}\bigg{]}=\frac{k_{\tau}\mu\omega^{2}r^{2}}{h}+\frac{ h^{3}}{12\mu}\bigg{[}G_{r}\bigg{(}\frac{\partial p}{\partial r}\bigg{)}^{2}+G_{ \theta}\bigg{(}\frac{1}{r}\frac{\partial p}{\partial r}\bigg{)}^{2}\bigg{]} \tag{4}\]  
where \(k_{\tau}\) is the shear turbulence coefficient. The boundary condition is as follows:  
\[\left\{ \begin{array}{ll}t_{i}=t_{in}\;i\in\Gamma_{4},\Gamma_{4}:\theta=0\\ \frac{\partial t_{i}}{\partial r}=0\;i\in\Gamma_{5},\Gamma_{5}:r=R_{1}\end{array} \right.\] 3. The viscosity-temperature equation  
The relationship between the dynamic viscosity and temperature of the water is as follows:  
\[\mu=A\text{e}^{B/(T+\mathbb{C})} \tag{5}\]  
where \(A\), \(B\), and \(C\) are constants and equal to 2.16 \(\times\) 10\({}^{-5}\) Pa\(\cdot\)s, 619.29 K, and 141.69 K, respectively. 4. The film thickness equation  
The film thickness equation of the WTTB considering the pad deformation is as follows:  
\[h=h_{z}+\gamma_{p}[r\sin(\theta_{p}-\theta)-r_{z}\sin(\theta_{p}-\theta_{z})] +\delta(r,\theta) \tag{6}\]  
where \(h_{z}\) is the film thickness at the fulcrum of the pad, \(\gamma_{p}\) is the swing angle of the pad, \(\theta_{p}\) is the position angle of the pitch line, \(r_{z}\) and \(\theta_{z}\) are the radial and circumferential positions of the fulcrum, respectively, and \(\delta\)(r,\(\theta\)) is the deformation of the pad. 5. The elastic deformation equation  
The [PERSON]'s modulus of the pad surface is small, and [PERSON]'s deformation model does not fit it well [21]. The elastic deformation equation of the pad surface can be defined as  
\[\delta(r,\theta)=\iint\limits_{\Omega}G(r,\theta,i,j)p(i,j)dsdt \tag{7}\]  
where \(G\)(r,\(\theta\),\(i\),\(j\)) is the pad surface's deformation coefficient produced by the water film pressure acting at the node, \(\Omega\) is the water film area, and \(\delta\)(r,\(\theta\)) is the deformation of the node (\(i\),\(j\)). In the three-dimensional model of the bearing, \(G\)(r,\(\theta\),\(i\),\(j\)) can be calculated using the Finite Element Method (FEM) when the element center node (\(i\),\(j\)) is subjected to a unit of pressure.