## 2 Numerical Method and Validation
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Label**} & \multicolumn{2}{c}{**Maximum Element Size**} & \multicolumn{2}{c}{**Minimum Element Size**} & \multicolumn{2}{c}{**Boundary Condition**} & \multirow{2}{*}{**Material**} & \multirow{2}{*}{**Geometry**} \\  & **(mm)** & & & & \\ \hline a & 0.5 & 0.04 & Pressure\_inlet & Fluid & Left edge \\ b & 0.48 & 0.038 & Pressure\_inlet & Fluid & Left edge \\ c & 0.45 & 0.035 & Pressure\_inlet & Fluid & Left edge \\ d & 0.25 & 0.015 & Pressure\_inlet & Fluid & Left edge \\ e & 0.52 & 0.042 & - & Fluid & - \\ f & 0.4 & 0.032 & - & Fluid & - \\ g & 0.25 & 0.015 & - & Fluid & - \\ h & 0.47 & 0.039 & Bristle\_Upper & Porous medium & Body \\ i & 0.45 & 0.035 & Bristle\_FenceHeight & Porous medium & Body \\ j & 0.3 & 0.02 & - & Fluid & - \\ k & 0.31 & 0.0254 & - & Fluid & - \\ l & 0.3 & 0.02 & - & Fluid & - \\ m & 0.45 & 0.035 & Pressure\_outlet & Fluid & Right edge \\ n & 0.45 & 0.035 & Pressure\_outlet & Fluid & Right edge \\ o & 0.3 & 0.02 & Pressure\_outlet & Fluid & Right edge \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Element-sizing method for each labeled mesh. Figure 2: (**a**) Computational domain and boundary conditions including porous medium model grid systems; (**b**) labeled meshes for brush seals without clearance used by (**c**) [PERSON] et al. [25] and (**d**) [PERSON] et al. [26], and for brush seals used by [PERSON] et al. [27] with (**e**) clearance = 0.27 mm and (**f**) clearance = 0.75 mm. ANSYS Fluent, the leakage flow rate was calculated at the pressure boundary inlet using Equation (1):  
\[\dot{m}=\int_{A}\rho\cdot\overset{\rightarrow}{u}\cdot dA. \tag{1}\]  
As shown in Figure 3, the difference in the predicted leakage flow rate between the two models was minimal. However, the k-\(\varepsilon\) model was selected for analyzing the flow field of the brush seal. This model is known to provide accurate predictions for high-Reynolds-number flows under high-pressure and high-speed operating conditions [35], and it has also been reported to yield reliable results when pressure boundary conditions are applied at the inlet and outlet [39]. Therefore, the realizable k-\(\varepsilon\) model, which is commonly used for brush seals, was adopted in this study, and enhanced wall treatment was applied to predict the near-wall flow more accurately [40]. Brush seals, which consist of densely packed bristles near rotating components, are particularly sensitive to wall effects on fluid flow. Enhanced wall treatment is a robust approach that combines low-Reynolds number models for resolving the near-wall region and wall functions for regions farther from the wall. This combination ensures a smooth transition between the viscosity-affected near-wall region and the fully turbulent outer region [41]. The mesh configurations presented in Figure 2 were designed to ensure that y + <5, which is required for applying enhanced wall treatment [42]. A grid-sensitivity test was conducted, the results of which are presented in Figure 4. The criterion for grid independence was determined by selecting the mesh size that resulted in a <0.2% error in leakage flow rate for all three geometries [25; 26; 27; 43]. For the [PERSON] et al. [25] geometry, grid independence was achieved with a mesh count of 62,750 or more; to minimize the error in leakage flow rate, 78,783 elements were selected. For the [PERSON] et al. [26] geometry, grid independence was confirmed at 113,340 elements, and the simulations were performed with 146,025 elements. For the [PERSON] et al. [27] geometry, with a 0.27 mm clearance, grid independence was achieved with 90,762 elements, and 110,020 elements were selected for the final simulations. The porosity of the bristle pack in the brush seal was calculated for the three geometries [25; 26; 27] using the formula developed by [PERSON] et al.