# The Investigation of a New End Wall Contouring Method for Axial Compressors
## 3 Application in a High-Load Compressor Cascade
### The Baseline Cascade and CFD Method  
The research object is a high-load linear compressor cascade with a designed flow turning angle of 52\({}^{\circ}\). The aspect ratio of the cascade is 2.0, and the solidity exceeds 2.1. There are detailed experimental studies that have been carried out on this cascade [26; 27], showing that severe corner separation arises in this cascade even under the design incidence (\(i=-1^{\circ}\)), and it will develop to corner stall when \(i\) exceeds 7\({}^{\circ}\). Thus, the working point with \(i=+7^{\circ}\) is named the near-stall (NS) point for brevity.  
The computational domain of the cascade only includes a half-span region of the cascade. Figure 6 provides an overview of the computational mesh. The inlet and outlet passages both use long extended \"H\" blocks. The blade-to-blade flow surface employs refined nodes in the center with the \"O4H\" topology to achieve higher orthogonality. The number of nodes in some critical sections are also presented. The computational domain extended to 1.2 times the axial chord (\(C_{\text{a}}\)) upstream of the LE, where the inlet velocity is taken during the experiment, and to 2.0 \(C_{\text{a}}\) downstream of the TE for the uniformity of the outflow parameter. The mesh in the boundary layer region upon the blade surface and the end wall is further refined for accuracy in calculating the corner separation. The growth rate of mesh from the solid wall to the flow field is less than 1.1, and the \(y^{+}\) is less than two at the first level upon the solid wall. The boundary condition is given according to the experimental results. The inlet boundary defines the static temperature and the inflow velocity. The thickness of the inlet boundary layer is about 12.5%_span_. The outlet boundary defines a uniform static pressure.  
The flow field of the cascade is regarded as compressible and full-turbulent. The \(Re\) for the current cascade is 1.4 \(\times\) 10\({}^{5}\). Table 2 provides details of the inflow condition. It is calculated with Numeca Fine/Turbo by solving the compressible steady Reynolds-averaged Navier-Stokes (RANS) equations with the k-\(\omega\) turbulence model. The spatial discretization scheme uses the second-order central scheme. The convergence criterion was set to a value of 1 \(\times\) 10\({}^{-6}\) for the RMS residual values. Local time stepping techniques were applied to speed-up convergence.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Inlet Conditions** & **Values** \\ \hline static temperature & 288.15 K \\ velocity of mainflow & 26.5 m/s \\ thickness of inlet boundary layer & 12.5\%_span_ \\ turbulent intensity & 0.3\% \\ \hline \hline \end{tabular}
\end{table}
Table 2: CFD settings for the inflow condition of the cascade.  
Figure 6: Overview of the computational mesh.  
All the above CFD settings are the same as previous numerical studies on this cascade [4], in which the CFD results were validated using the experimental outflow parameter and the oil-flow test on the SS and end wall surface. A grid convergence study was also applied, showing that a total of 0.75 million grid nodes is sufficient. Here, we offer only the validation of outflow loss coefficient (_Loss_) and flow angle (\(\beta\)) under the design working condition (\(i=-1^{\circ}\)), as in Figure 7. The horizontal axis shows the loss coefficient, defined as  
\[Loss=\frac{P^{*}-P^{*}_{\text{in}}}{P^{*}_{\text{in}}-P_{\text{in}}} \tag{9}\]  
where \(P^{*}_{\text{in}}\) and \(P_{\text{in}}\) represent the inlet total pressure and static pressure in the mid-span position. The vertical axis means spanwise position normalized using the height of the blade. The result labeled \"TB\" means the working condition with a thicker inlet boundary layer and intensified corner separation. The comparison between the CFD and experimental results shows that the numerical method achieves good accuracy in calculating secondary flow and loss increase when intensified corner separation occurs.