## 2 Methodology
### Numerical Methods  
#### 2.2.1 Governing Equations  
The Reynolds-average Navier-Stokes (RANS) method is used to numerically simulate the three-dimensional incompressible turbulence of the water turbine, ignoring the influence of temperature and using water as the flow medium. The continuity equation is then:  
\[\frac{\partial\rho}{\partial t}+\frac{\partial}{\partial x_{j}}\big{(}\rho u_{j} \big{)}=0 \tag{1}\]  
where \(\rho\) is the density, \(u_{\rm j}\) is the velocity component of the fluid, and the momentum equation becomes:  
\[\frac{\partial}{\partial x_{j}}\Big{(}\rho u_{i}u_{j}\Big{)}=-\frac{\partial p }{\partial x_{i}}+\frac{\partial}{\partial x_{j}}\left[\mu\left(\frac{\partial u _{i}}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\right)\right]+ \frac{\partial}{\partial x_{j}}\Big{(}-\rho\overline{u_{i}^{\prime}u_{j}^{ \prime}}\Big{)}+F_{i} \tag{2}\]  
where \(\rho\) is the fluid density, \(u_{i}\) and \(u_{j}\) are the three-dimensional velocity components of the fluid, \(p\) is static pressure, \(\mu\) is the dynamic viscosity, and \(F_{i}\) is the external source term in the \(i\) direction. The shear stress transport (SST) model based on two-equation \(k\)-\(\omega\) is used to simulate the turbulent flow field in the water turbine. If there is a reverse pressure gradient in the flow, this model predicts the flow field more accurately by limiting the generation of eddy viscosity and turbulent kinetic energy [33]. The formation of vortices in the water turbine is related to the flow separation under the reverse pressure gradient. Therefore, it is believed that this model is suitable for the modeling of this study, and its transport equation is as follows:  
\[\frac{\partial}{\partial t}\big{(}\rho k\big{)}+\frac{\partial}{\partial x_{i }}\big{(}\rho u_{i}k\big{)}=\widetilde{P}_{k}-\beta^{*}\rho k\omega+\frac{ \partial}{\partial x_{i}}\bigg{[}\big{(}\mu+\sigma_{k}\mu_{t}\big{)}\frac{ \partial k}{\partial x_{i}}\bigg{]} \tag{3}\]  
\[\frac{\partial}{\partial t}\big{(}\rho\omega\big{)}+\frac{\partial}{\partial x _{i}}\big{(}\rho u_{i}\omega\big{)} \tag{4}\]  
\[v_{t}=\frac{a_{1}k}{\max(a_{1}\omega,S_{E})} \tag{5}\]  
\[P_{k}=\mu_{t}\frac{\partial u_{i}}{\partial x_{j}}\Bigg{(}\frac{\partial u_{i }}{\partial x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\Bigg{)}\rightarrow \widetilde{P}_{k}=\min(P_{k},10\cdot\beta^{*}\rho k\omega) \tag{6}\]  
\[F_{1}=\tanh\Bigg{\{}\left\{\min\bigg{[}\max\bigg{(}\frac{\sqrt{k}}{\beta^{*} \omega y^{*}},\frac{500v}{y^{2}\omega}\bigg{)},\frac{4\rho\sigma_{\omega 2}k}{CD_{k\omega}y^{2}}\bigg{]}\right\}^{4}\Bigg{\}} \tag{7}\]  
\[F_{2}=\tanh\Bigg{\{}\left[\max\bigg{(}2\frac{\sqrt{k}}{\beta^{*}\omega y^{*}},\frac{500v}{y^{2}\omega}\bigg{)}\right]^{2}\Bigg{\}} \tag{8}\]  
\[CD_{k\omega}=\max\bigg{(}2\rho\sigma_{\omega 2}\frac{1}{\omega}\frac{\partial k}{ \partial x_{i}}\frac{\partial k}{\partial x_{i}},10^{-10}\bigg{)} \tag{9}\]  
where \(k\) is turbulent kinetic energy, \(\omega\) is turbulent dissipation rate, \(S\) is the strain rate magnitude, \(y\) is the distance to the nearest wall, \(v_{\rm t}\) is the eddy viscosity coefficient, \(\beta^{*}=0.09\), \(\alpha_{1}=5/9\), \(\beta_{1}=0.075\), \(\sigma_{k1}=0.85\), \(\sigma_{\omega 1}=0.5\), \(\alpha_{2}=0 mesh element is above 0.3. The calculation domain includes inlet pipe, inlet nozzle, runner, and outlet pipe. The runner is defined as the rotational component, while the inlet pipe, inlet nozzle, and outlet pipe are considered stationary. The Frozen Rotor interface was used in the connection interface between the runner and the runner hosing. The inlet and outlet boundary conditions are set to the normal boundary velocity and the average static pressure, respectively. The inlet velocity is set to the corresponding velocity under the design flow rate, and the outlet static pressure is set to the same value as the actual operation. All wall boundaries adopt non-slip wall conditions. The General Grid Interface (GGI) interface grid processing method is adopted.