## 3 Mathematics & Materials  
In these cases, each hole angle with different coolant injection orientations are confirmed in the former researches, so the coolant inlet orientation is orthogonal to the wall. According to the working condition of one F model gas turbine, the temperature and the mass flux rate of the mainstream flow is set as 1300 K and 32.72 kg/s, respectively. The turbulence intensity is set at 5%, which can be estimated based on the mass flow rate, area, hydraulic diameter of the gas inlet, and gas viscosity. In order to reduce calculation, the number of impact hole was reduced to only three in the coolant chamber. Therefore, the value of pressure on the jet hole inlet is used as the initial conditions, which is set at 1.821 MPa. The pressure recovery coefficient is given as 0.95, which means that the ratio of pressure between the inlet and outlet of transition piece with smooth surface. The turbulent intensity on the coolant inlet is 10%. Details of boundary conditions are ascertained in Table 1[13]. In the mainstream chamber, it is assumed that the main streamflow is a mixture of N\({}_{2}\), O\({}_{2}\), H\({}_{2}\)O, and CO\({}_{2}\), as well as rare gases. In another chamber, air as the cooling flow is used for all of the simulations. The material of the thermal surface is using Nimonic 263 (Hucheng industry (Shanghai) Co., Ltd., Shanghai, China), for which information could be found from internet. The model is a computational domain that is made up by hexahedron meshes in the software, ANSYS-ICEM, version 18.0. In Figure 3, the grid sensitivity test for the simplified two-chamber model with the biomimetic thermal surface are carried out. When the number of cells increase from 2,227,680 to 2,962,080, the area weighted average temperature is declined by 0.3%. The calculation deviation of temperature is insignificant on further increasing the numbers of cells. Thus, 2,200,000 cells are used as a grid independent mesh for obtaining the solution variables in our further simulation. In order to intuitively acquire the quantity of heat that is taken away in the process of heat exchange in the cooling chamber, the flow rate temperature \(\lambda\), which is provided as an indicator that is computed by the product of flow rate and temperature, and it is defined as  
\[\lambda=\int T\rho\overrightarrow{v}\cdot d\overrightarrow{A}=\sum_{i=1}^{n}T _{i}\rho_{i}\overrightarrow{v}_{i}\cdot\overrightarrow{A}_{i} \tag{1}\]  
where \(\rho\) is the density of the coolant and \(\overrightarrow{v}\) is the facet velocity on the selected field. Also, the heat transfer coefficient can be defined as  
\[\eta=\frac{\beta_{i}-\beta_{s}}{\beta_{s}} \tag{2}\] \begin{table}
\begin{tabular}{c c c} \hline \hline
**Component** & **Boundary Conditions** & **Magnitude** \\ \hline \multirow{4}{*}{Mainstream inlet} & Mass flux rate & 32.72 kg/s \\  & Gas temperature & 1300 K \\  & Turbulent intensity & 5\% \\  & Hydraulic diameter & 0.324 m \\ \hline \multirow{4}{*}{Mainstream outlet} & Pressure & 1.573 MPa \\  & Turbulent intensity & 5\% \\  & Hydraulic diameter & 0.324 m \\  & Convection coefficient & 10 W/m\({}^{2}\)K \\ \hline \multirow{4}{*}{Coolant chamber} & Air temperature & 300 K \\  & Pressure & 1.821 MPa \\ \cline{1-1}  & Pressure recovery coefficient & 0.95 \\ \cline{1-1}  & Turbulent intensity & 10\% \\ \cline{1-1}  & Hydraulic diameter & 0.01026 m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Boundary conditions [13]. which serves as an indicator to value the performance of impinging cooling. In Equation (2), \(i\) is the number of group and \(s\) represents the group of smooth thermal surface. Also, \(\beta\) is defined as  
\[\beta=\lambda_{out}-\lambda_{in} \tag{3}\]  
In Equation (3), \(\lambda_{out}\) and \(\lambda_{in}\) are the flow rate temperature on the outlet and the inlet surfaces.