## 2 Research Object and Numerical Analysis Model
### Numerical Method  
The phenomenon of fluid flow and heat transfer satisfies certain physical laws, and its governing equation is a mathematical description of the basic physical rules. The equations are solved by numerical calculating method to explore the essential problems inside the flow and heat transfer phenomenon, so as to provide the theoretical basis for guiding the application of engineering practice. The solved equations include mass equation, momentum equation, and energy equation as shown below. The fluid state equation is also supplemented in order to satisfy the solution.  
\[\frac{\partial\rho u_{i}}{\partial x_{i}}=0 \tag{9}\]  
\[\frac{\partial\rho u_{i}u_{j}}{\partial x_{i}}=\rho g_{i}+F_{i}-\frac{\partial P }{\partial x_{i}}+\frac{\partial}{\partial x_{i}}\left(2\mu S_{ij}\right) \tag{10}\]  
\[\frac{\partial\rho u_{i}E_{0}}{\partial x_{i}}=\rho u_{i}F_{i}-\frac{\partial q _{i}}{\partial x_{i}}+\frac{\partial}{\partial x_{j}}\left(u_{i}T_{ij}\right) \tag{11}\]  
\[\rho=f(p,T) \tag{12}\]  
where \(F_{i}\) is the body force, \(P\) is the pressure, \(S_{ij}\) is the strain rate tensor, \(E_{0}\) is the total internal energy, and \(T_{ij}\) is the surface force.  
In this research, the RANS method is adopted and solved by the commercial finite-volume solver. [PERSON] et al. [28] experimentally and numerically verified a three-pass channel arranged turning vanes and ribs. According to their results, when the SST \(k\)-\(\omega\) turbulence model is chosen to simulate, the maximum deviation is 6.5% compared to the experiment. Meanwhile, the errors based on Realizable \(k\)-\(\omega\) model and RNG \(k\)-\(\omega\) model are 19.3% and 22.4%, respectively. Thus, in this paper, the SST \(k\)-\(\omega\) model is selected for further study.  
The detailed boundary conditions are shown In Table 3. An invariable heat flux density 5000 W\(\cdot\)m\({}^{2}\) is given on PS, SS, and tip wall surface. The other surfaces are adiabatic non-slip boundaries. Since \(Re\) is roughly proportional to the inlet flow rate, the inlet is given a constant mass flow rate calculated by \(Re\). The channel inlet temperature of fluid is 298.15 K. The turbulence intensity is 5%. The outlet pressure of the channel pressure is set at 1 atm. The cooling medium in the channel is the ideal air gas. The rotational speed is calculated by \(Ro\). The high-resolution scheme is adopted to improve the calculation accuracy. When the residuals are reduced to \(10^{-6}\), the calculation is judged to be convergence.  
The overall and detailed grid of the case4 channel is shown in Figure 3. Since the boundary layer will be formed close to the surface during the flow of the working substance,  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Value** \\ \hline Heat flux density on PS, SS, and tip wall & 5000 [W\(\cdot\)m\({}^{2}\)] \\ surface & 298.15 [K] \\ Inlet temperature & 5\% \\ Outlet pressure & 1 [atm] \\ \hline \hline \end{tabular}
\end{table}
Table 3: The detailed boundary conditions.  
the near-wall grid of the channel and the turning vane is refined. The O-type grid is used at the protrusion structure to improve the grid quality and ensure \(y^{+}<1\).  
The mesh independence verification is carried out in stationary and rotating states based on the case4 model. Grids with different densities have been established, and the quantity of elements is 0.79 million, 1.44 million, 2.33 million, and 4.40 million, respectively. Table 4 shows the results under each grid. It can be seen form the table that the relative deviation of the \(Nu\) on tip wall (\(Nu_{\text{up}}\)) and pressure drop (\(\Delta p\)) based on grid3 has been reduced to less than 1%, which is sufficient to meet the accuracy requirements. Therefore, the subsequent calculations are based on grid 3.