# Flow Characteristics and Loss Models of the Transonic Radial Turbine
## 2 Numerical Model  
A radial inflow valenless turbine was designed. Table 1 shows the detailed geometry parameters of the turbine. The NUMECA software system is used to study the performance and aerodynamics of the turbine in design and off-design conditions, and the structured grid is produced by the NUMECA Autogrid. Figure 2 shows the 3D geometry and meridional cross-section of the turbine. The main geometry parameters are shown in Figure 2b. In order to improve computational efficiency and reduce the number of computational grid cells, a single passage simulation was performed instead of the entire turbine simulation. Figure 3 shows a single passage computing grid. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Geometry Parameters** & **Turbine** & **Units** \\ \hline Inlet radius & 48.16 & mm \\ Inlet blade height & 10.01 & mm \\ Exit tip radius & 41.78 & mm \\ Exit hub radius & 16.02 & mm \\ Inlet blade angle & 0 & degree \\ Exit blade angle & –58 & degree \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Detailed geometry of turbine. Figure 3: The computational grids of the single passage. Figure 2: The 3D geometry and meridional cross-section of turbine. (**a**) The 3D geometry of turbine. (**b**) Meridional cross-section of turbine. Before the CFD analysis, a grid study was conducted to check the sensitivity of the grid to the results. The number of the grid has a very important influence on the result of the calculation. If the number is too large, it will waste a lot of time in the calculation. If it is too small, the results of the calculation will be not accurate. So, it is necessary to study the influence of the gridding number on the performance of the turbine. Six types of mesh with different numbers of grid points were generated to calculate the model. The topology of the mesh and the boundary conditions were set the same. Figure 4 shows the effect of gridding numbers on turbine efficiency. It shows that as the number of grids increases, the efficiency of the turbine also increases, but when the number of grids exceeds 700,000, the efficiency remains almost unchanged. Therefore, the final grid number of the rotor grid is close to 700,000, which is higher than the minimum requirement indicated by the grid independence study. In order to effectively capture complex 3D flow patterns in the flow channel, HO topology is used in the rotor channel. The mesh topology of the skin blocks around the rotor blades is O topology, and H topology is used in the inlet, outlet and mainstream channel domains. The grid is refined at the near wall, leading edge and trailing edge areas. The rotor blade tip clearance is 0.4 mm. The gap area is gridded by 17 grid points in the span direction. The minimum skew angle of the grid cells is 19. An S-A turbulence model was used as the turbulence model to achieve computational efficiency and good accuracy. For the separated flow, the prediction of the S-A model is better than that of the algebraic model, and the differential equation will not have serious numerical difficulties. Studies have shown that the S-A model is an excellent engineering tool for predicting turbulence characteristics. NUMECA's solver is FINE/Turbo, which is based on the element-centric finite volume method and is associated with [PERSON]'s central scheme as spatial discretization and the explicit Runge-Kutta time integration method [13]. The mass flow, velocity direction and temperature are defined at the inlet boundary. The static pressure is defined at the outlet boundary. In order to verify the accuracy of the simulation results, the experimental results were compared with the simulation results. Figure 5 shows the comparison of the total to static efficiency (the ratio of the actual work output and the theoretical work output based on the rotor pressure drop) between the simulation results and the experimental results.