## 2 Methodology of the Numerical Simulations  
For the unsteady turbulent flow in hydraulic Francis turbines, the differential form of the Navier-Stokes equation and the continuity equation can be expressed in [PERSON] notation in Cartesian coordinates as  
\[\frac{\partial(\rho v_{i})}{\partial t}+\frac{\partial(\rho v_{i}v_{j})}{ \partial x_{j}}=-\frac{1}{\rho}\frac{\partial p}{\partial x_{i}}+\frac{ \partial\tau_{ij}}{\partial x_{j}}+\rho f_{i} \tag{1}\]  
\[\frac{\partial\rho}{\partial t}+\frac{\partial(\rho v_{i})}{\partial x_{i}}=0 \tag{2}\]  
where \(\rho\), \(v_{i}\), \(v_{i}\), \(p\), \(\tau_{ij}\), and \(f_{i}\) are the fluid density, fluid velocity components, fluid pressure, surface force tensor, and mass force source term, respectively. The \(i\) and \(j\) indices are the three-dimensional coordinate directions. The water inside the hydraulic Francis turbines can be considered to be at a constant temperature and incompressible, and the above equations can be written as  
\[\frac{\partial v_{i}}{\partial t}+\frac{\partial(v_{i}v_{j})}{\partial x_{j}} =-\frac{1}{\rho}\frac{\partial p}{\partial x_{i}}+\frac{\partial}{ \partial x_{j}}\left[v(\frac{\partial v_{i}}{\partial x_{j}}+\frac{\partial u _{j}}{\partial x_{i}})\right]+f_{i} \tag{3}\]  
\[\frac{\partial v_{i}}{\partial x_{i}}=0 \tag{4}\]  
where \(\
u\) is the kinematic viscosity. The flow inside the hydraulic Francis turbines is turbulent, with characteristics of unsteady-state, nonlinear, and multi-scale. It is currently impractical to solve the Navier-Stokes equations directly numerically for complex engineering problems. For Francis turbines, engineering design focuses more on the pressure distribution and flow pattern of the flow field rather than all the turbulent details, so the turbulence in the above equations can be appropriately simplified. Reynolds-averaged Navier-Stokes (RANS) is the most commonly used method, which decomposes each quantity in turbulence into the sum of the time average and the pulsating value. Any instantaneous variable \(\Theta\) of the turbulent flow can be decomposed into the time average \(\overline{\Theta}\) and the pulsating value \(\Theta^{\prime}\):  
\[\Theta=\overline{\Theta}+\Theta^{\prime} \tag{5}\]  
Since the average of the pulsation value is 0, Equations (3) and (4) can be written in the format of Reynolds-averaged Navier-Stokes equations (RANS equations):  
\[\frac{\partial\overline{v_{i}}}{\partial t}+\frac{\partial(\overline{v_{i}v_{ j}})}{\partial x_{j}}=-\frac{1}{\rho}\frac{\partial\overline{p}}{\partial x_{i}}+ \frac{\partial}{\partial x_{j}}[v(\frac{\partial\overline{v_{i}}}{\partial x _{j}}+\frac{\partial\overline{v_{j}}}{\partial x_{i}})]-\frac{\partial \overline{v_{i}^{\prime}v_{j}^{\prime}}}{\partial x_{i}}+f_{i} \tag{6}\]  
\[\frac{\partial(\overline{v_{i}})}{\partial x_{i}}=0 \tag{7}\]  
The RANS equations introduce a new unknown quantity, the Reynolds stress \(\overline{v_{i}^{\prime}v_{j}^{\prime}}\)[33]. In order to close the set of equations, a variety of turbulence models with the following parameters that link the Reynolds stress with the average turbulence quantity have been introduced [34; 35]:  
\[\omega=\frac{\varepsilon}{k} \tag{8}\]  
\[\varepsilon=\
u\frac{\overline{\delta v_{i}^{\prime}}}{\overline{\delta x_{k}}} \frac{\overline{\delta v_{i}^{\prime}}}{\overline{\delta x_{k}}} \tag{9}\]  
\[k=\frac{1}{2}\overline{v_{i}^{\prime}\overline{v}_{j}} \tag{10}\]  
where \(\omega,k,\) and \(\varepsilon\) are the turbulent kinetic energy dissipation rate, the average turbulent kinetic energy, and the turbulent frequency, respectively. For turbulent flow in hydraulic Francis turbines, the eddy viscosity model is the most widely used model.