# Numerical Study of Endwall Modification with Micro-Scale Ribs in a Turbine Cascade
## 1 Introduction
They found that the cooling characteristic of the endwall could be effectively improved by the small-scale structures. [PERSON] et al. [20] carried out a numerical investigation on the flow and cooling characteristics of a micro-ribbed vane endwall. Their findings indicated that coolant migration was inhibited by the ribs and benefited the lateral coverage of the coolant. In addition, another endwall modification was carried out by [PERSON] et al. [21], indicating that the filled passage performed better than the original case with respect to aerodynamic and cooling performance. So far, several methods for endwall modification have been proposed, aiming to enhance certain characteristics. Further information about the benefits and demerits of endwall modification is listed in Table 1. As highlighted from the above studies, endwall modifications have been investigated by researchers, but most of them only focused on several aspects instead of a comprehensive evaluation of a micro-scale rib on an endwall. Therefore, to gain a comprehensive understanding of the effect of micro-scale ribs on film cooling (including phantom cooling), heat transfer, and aerodynamic characteristics, original and ribbed endwalls are studied with upstream leakage and various rib types and rib spacings in this paper. A detailed flow pattern for the endwall and ribs is given. This paper is expected to provide extensive modification advice for the cooling design of turbine endwalls. \begin{table}
\begin{tabular}{c c c c} \hline  & \multicolumn{2}{c}{**Benefits**} & \multicolumn{2}{c}{**Demerits**} \\ \hline \multirow{3}{*}{Contoured endwall} & \(\bullet\) & Reduce flow loss & \(\bullet\) & The effect of loss reduction is not good as non-axisymmetric endwall \\  & \(\bullet\) & Improve the use of leakage & & good as non-axisymmetric endwall \\ \hline \end{tabular}
\end{table} 
Table 1: Information for endwall modification. Figure 3 shows the numerical model of ribbed endwall for upstream slot leakage. The numerical model of cascade channel refers to the experimental facilities, which indicated that the sizes of the cascade channel and micro-scale rib were consistent with the experimental conditions. The detailed information of experimental facilities and the measurement method is the same as in the literature [14]. When simulating the cooling characteristics of the upstream slot for the ribbed endwall, the double sides of the passage were given as periodic boundary conditions, and the remaining walls were given as adiabatic non-slip walls. The mainstream inlet was given as velocity boundary, the mainstream outlet was given as pressure outlet, and the coolant inlet was given as mass flow inlet. RANS equation was solved using the commercial software CFX 19.0. The turbulence term was discretized with high resolution and the RMS residual was less than \(10^{-5}\) to assume convergence. To meet the actual requirement of density ratio, CO\({}_{2}\) was used for coolant and the air was used for mainstream. The solved governing equations are shown in Equations (1)-(3). Continuity equation:  
\[\frac{\partial\rho}{\partial t}+\
abla\cdot\left(\rho\overset{\rightarrow}{U} \right)=0 \tag{1}\]  
where \(\rho\) represents the fluid density and \(\overset{\rightarrow}{U}\) represents the velocity vector of fluid. Figure 3: Computational model (pink: blade; orange: endwall; gray: hub). Figure 2: Schematic of original endwall and ribbed endwall with different rib types. (**a**) Original; (**b**) Horizontal rib; (**c**) Tilted rib; (**d**) Vertical rib. Momentum equation:  
\[\rho\frac{\partial\overset{\rightarrow}{u}}{\partial t}+\rho\left(\overset{ \rightarrow}{u}\cdot\
abla\right)\overset{\rightarrow}{u}=\rho\overset{ \rightarrow}{f}+\
abla\cdot\sum \tag{2}\]  
where \(\overset{\rightarrow}{f}\) and \(\sum\) represent the mass force and surface force acting on the fluid surface, respectively.