## 2 Methodology
### Numerical Simulation of HTB of Star Wheel Shift in WSSC  
#### 2.2.1 Governing Equations  
A large amount of water injected and a low temperature increase in lubricating water during the working process of WSSCs will cause the energy equation to be disregarded [24]. Water flow in HTBs is governed by the following equations. According to the law of conservation of mass, the equation of mass conservation (i.e., continuity equation) can be written as follows:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial\rho u}{\partial x}+\frac{ \partial\rho v}{\partial y}+\frac{\partial\rho w}{\partial z}=0 \tag{4}\]  
The law of conservation of momentum (i.e., Navier-Stokes equation) is shown as follows:  
\[\left\{ \begin{array}{l}\rho\Big{(}\frac{\partial u}{\partial t}+u\frac{ \partial u}{\partial x}+v\frac{\partial u}{\partial y}+w\frac{\partial u}{ \partial z}\Big{)}=\rho X-\frac{\partial p}{\partial x}+\mu\Big{(}\frac{ \partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}+\frac{ \partial^{2}u}{\partial z^{2}}\Big{)}\\ \rho\Big{(}\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v \frac{\partial v}{\partial y}+w\frac{\partial v}{\partial z}\Big{)}=\rho Y- \frac{\partial p}{\partial y}+\mu\Big{(}\frac{\partial^{2}v}{\partial x^{2}} +\frac{\partial^{2}v}{\partial y^{2}}+\frac{\partial^{2}v}{\partial z^{2}} \Big{)}\\ \rho\Big{(}\frac{\partial v}{\partial t}+u\frac{\partial w}{\partial x}+v \frac{\partial w}{\partial y}+w\frac{\partial w}{\partial z}\Big{)}=\rho Z- \frac{\partial p}{\partial z}+\mu\Big{(}\frac{\partial^{2}w}{\partial x^{2}} +\frac{\partial^{2}w}{\partial y^{2}}+\frac{\partial^{2}w}{\partial z^{2}} \Big{)}\end{array} \right. \tag{5}\]  
Figure 5: Variation of gas force on star wheel with rotation angle of star wheel. \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Value** \\ \hline Internal diameter of the HTB/D\({}_{1}\) & 55 mm \\ Internal diameter of the radial groove/D\({}_{2}\) & 72 mm \\ Internal diameter of the annular groove/D\({}_{3}\) & 89 mm \\ External diameter of the annular groove/D\({}_{4}\) & 95 mm \\ External diameter of the radial groove/D\({}_{5}\) & 111 mm \\ External diameter of the HTB/D\({}_{6}\) & 120 mm \\ Width of the groove/e & 3 mm \\ \hline \end{tabular}
\end{table} 
Table 2: Parameters of the HTB for the WSSC prototype. where \(\rho\) and \(\mu\) are the density and dynamic viscosity, respectively, of water; \(p\) is the water pressure in bearing clearance; \(u\),\(v\), and \(w\) are the velocity components in the \(x\), \(y\), and \(z\) directions, respectively; and \(X\),\(Y\), and \(Z\) are the external forces per unit mass of lubricating water. Whether a fluid is in a laminar or turbulent state can be judged using the Reynolds equation [25]:  
\[Re=\frac{\rho vl}{\mu} \tag{6}\]  
Note that 2300 is the critical Reynolds number of flow for liquid transition from laminar flow to turbulent flow [25]. In the current study, the Reynolds number of an HTB is expressed as follows:  
\[Re=\frac{\rho\cdot 2\pi nR_{0}\cdot h}{60\mu} \tag{7}\]  
where \(\rho=10^{3}\) kg/m\({}^{3}\), the pitch diameter of the HTB \(R_{0}=45.05\) mm, \(\mu=10^{-3}\) Pa \(\cdot\) s, and the rotation speed of HTB \(n=2970\) r/min. All Reynolds numbers obtained by substituting the water film thickness to Formula 13 are below 2300. Thus, laminar flow is used for simulation. The following assumptions are necessary to construct the governing equations applicable to the hydrostatic water film of star wheel HTB. 1. Given that water does not slip on bearing surfaces, the speed of the water film and that of the HTB ring is the same at their interface. 2. Water film flowing in the bearing clearance is laminar without eddy current and turbulence. Hence, \(\partial p/\partial z=0\). 3. The volume force of water is negligible compared with the viscous force. That is, the external forces X, Y, and \(Z\) acting on the water film are 0. 4.