## 2 Research Object and Methodology  
The studied 1000 MW Francis turbine and its unit layout are depicted in Figure 1, and Table 1 lists the parameters of the unit. To analyze the flow characteristics, we constructed a three-dimensional model encompassing the entire flow channel of the unit (Figure 2), which was then subjected to meshing. The boundary conditions were established based on measured data, and we conducted flow calculations for the runner to evaluate the impact of radial deviation clearances on the turbine unit's flow characteristics. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Rated power & 1000 MW \\ Stay vane number & 23 \\ Guide vane number & 24 \\ Runner blade number & 15 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The parameters of the 1000 MW Francis turbine unit. Figure 1: The flow channel of the 1000 MW Francis turbine (**left**) and a cross-sectional view (**right**). Figure 2: Mesh of the entire flow domains. To ensure clarity for subsequent understanding, an explanation is provided here. For reasons of confidentiality and other considerations, certain calculation data were subjected to correlation or normalization processing. The formulas for this are organized as follows:  
\[D_{r}=\frac{D_{s}}{D_{e}} \tag{1}\]  
where \(D_{r}\) represents the normalized or relativized calculation results, \(D_{s}\) represents the computed simulated experimental data, and \(D_{e}\) represents the data obtained from actual engineering. To ascertain mesh independence, we generated meshes for the entire flow domains in Figure 3 while applying the boundary conditions corresponding to rated working conditions. As indicated in Table 2, there was no significant difference in efficiency between the two cases. Consequently, we chose to utilize the mesh with 5 million elements for subsequent calculations to reduce the computation time. For a more comprehensive validation of grid independence and parameter comparison, we refer readers to another article [30] authored by our team, which specifically validates grid numbers under unbiased ideal conditions. The radial deviation grid examined in this article is built upon this foundation, ensuring that the total number of deviation grids, after adjustment, closely aligns with the total number of grids in the unbiased ideal model. The element counts in each flow domain for the 5 million element mesh are detailed in Table 3. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Rated Conditions** & **Relative Head (\%)** & **Rated Opening (\%)** & **Relative Efficiency (\%)** \\ \hline Mesh1 (around & & & \\
5 million elements) & 100 & 100 & 99.6 \\ Mesh2 (around & & & \\
10 million elements) & 100 & 100 & 100 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mesh independence test. Figure 3: Mesh independence test. For the numerical simulations, we utilized the Reynolds time-averaging (RANS) Equation (4), which are built upon the Navier-Stokes (N-S) Equations (2) and (3). Our chosen turbulence model was the SST \(k\)-\(\omega\) turbulence model, described by Equation (5). Using the finite volume method, we discretized and solved the three-dimensional flow domains of the unit to obtain the turbulent flow field within it:  
\[\frac{\partial v_{i}}{\partial x_{i}}=0 \tag{2}\]  
\[\frac{\partial(\rho v_{i})}{\partial t}+\frac{\partial\big{(}\rho v_{i}v_{j} \big{)}}{\partial x_{j}}=\rho F_{i}+\frac{\partial}{\partial x_{i}}\left(\mu \frac{\partial v_{i}}{\partial x_{j}}\right)-\frac{\partial p}{\partial x_{i}} \tag{3}\]  
In the equations, the variables are defined as follows: \(v_{n}\) (with \(n\) representing \(i\), \(j\), \(k\) in Cartesian coordinates) for the three velocity direction components, \(x_{n}\) for the coordinates, \(f_{n}\) for the body force, \(\mu\) for the dynamic viscosity, \(t\) for the time, \(p\) for the pressure, and \(\rho\) for the water density.