## 2 Methodology
These schemes are the integral components of Total Variation Diminishing (TVD) algorithms and have demonstrated their accuracy in handling sharp gradients and intricate flow phenomena [13,14]. For the transonic/supersonic simulations involving steep gradients, such as shock waves or turbulent boundary layer, numerical schemes capable of providing accurate results, along with a good balance between stability and accuracy, are essential. Therefore, in addition to the numerical schemes, other factors such as turbulence modeling and specific software settings play a crucial role in determining the accuracy of the solution. The mesh density for the idealized periodic passage consisted of approximately 1.5 million nodes. This configuration was established based on earlier research [29], which identified the optimal balance between resolution and numerical stability and convergence time for studies related to turbomachinery. To ensure an accurate solution for the examined problem, a grid independence study was conducted for the entire cascade. This process involved six different grid resolutions, ranging from coarse to fine meshes. All generated grids are composed of unstructured tetrahedral cells with low skewness. The coarse meshes (7-, 11-, and 17-million-cell mesh) have refinements around the airfoil and in the wake, achieving a y+ distribution close to 1. This variation was achieved through modifications of the number of layers in the radial and axial directions, as well as the overall sizing of the geometry. For the next set (24, 30, and 36 million cells), an additional level of refinement was applied in the blades' test area to enhance mesh density, thus ensuring an accurate representation of the shockwave structures. The transition from a denser grid to a far-field grid was executed gradually, avoiding sudden transitions between adjacent cells. Also, the number of layers from the inflation layer was increased to assure proper definition of the boundary layer. For each grid resolution, the performance was assessed using CFD simulations, and the optimal grid was determined by comparing the results of these analyses. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Symbol** & **Parameters** & **Main Blade/Splitter Blade** \\ \hline c & Blade chord length [mm] & 71.16 & 45.22 \\ s & Pitch width [mm] & & 52 \\
1 & Blade span [mm] & & 10 \\ \(\sigma\) & Solidity & 1.368 & 0.869 \\ \(\gamma\) & Stagger angle [\({}^{\circ}\)] & 51.55 & 49.23 \\  & Inlet total pressure [Pa] & & 200,000 \\  & Inlet temperature [K] & & 293 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Cascade parameters and flow conditions. Turbulence was modeled using the k-\(\upomega\) SST (Shear Stress Transport) formulation with streamline curvature acceleration corrections [30]. The SST turbulence model was chosen due to its extensive applicability in simulating turbomachinery flows, allowing accurate performance predictions across a wide range of operating conditions [31; 32]. This two-equation turbulence model leverages blending functions to utilize the two underlying turbulence models that define it. It employs the k-epsilon model in the far field and transitions to the k-omega model near the walls. For the studied case, it was crucial to accurately capture both the flow structure near the blade walls and in the far field to determine the structure of shock waves. The computation for the full transonic cascade was performed for an imposed mass flow (0.4 kg/s) and total temperature (293 K) at the inlet patch, while at the outlet, atmospheric static pressure was assumed. The endwalls, as well as the main and splitter vanes, were modeled as adiabatic walls with no slip and without surface roughness (machined surface). For this simulation, the mass flow rate for each channel was extracted and served as the input boundary condition (i.e., inlet mass flow rate) for the single-passage simulations (as presented in Table 2). To assure the accuracy and stability of the solution, different convergence criteria were evaluated.