# Flow Control of Radial Inlet Chamber and Downstream Effects on a Centrifugal Compressor Stage
## 2 Methodology
### Verification and Validation  
#### 2.3.1 Grid Convergence Study  
Structured multi-block grids were generated for all the components using NUMECA IGG, and particular attention was paid to the grid refinements close to the solid walls. A grid convergence study was performed for all the CFD models to verify the grid independence of the numerical results in the present research. Here, the flow analysis model of the radial inlet chamber with eight guide vanes was provided as an instance for the comparison of the numerical results obtained with different grid resolutions. In order to verify grid independence, seven groups of computational grids were generated for this numerical model, and the changes in grid elements in each component followed the grid convergence method (GCI) provided by ASME (American Society of Mechanical Engineers) [38; 39]. Detailed grid distributions of the computational components in different schemes are listed in Table 2.  
Figure 2: Computational models for flow analysis and the downstream effects of the radial inlet chamber. As the grid number changed, the variations of the major flow parameters (e.g., the pressure, temperature and velocity) on the main sections were monitored and compared. The performance parameters were also analyzed for the CFD model. As a stationary component, the flow efficiency \(\eta_{ic}\) and total pressure loss coefficient \(\xi_{ic}\) of the radial inlet chamber are expressed as  
\[\eta_{ic}=\frac{\kappa}{\kappa-1}\cdot\frac{\ln(T_{s,0}/T_{s,in})}{\ln(p_{s,0}/ p_{s,in})} \tag{3}\]  
\[\xi_{ic}=(p_{t,in}-p_{t,0})/(\frac{1}{2}\rho_{in}V_{in}^{2}) \tag{4}\]  
Figure 3 shows the trends of the outlet static pressure and inlet chamber efficiency under the seven grid resolutions, where the flow parameters were extracted from the mass-averaged values in the corresponding sections. It can be clearly seen that when the grid number was less than a certain limit (e.g., \(7.5\times 10^{6}\) for this particular case), both the static pressure at the model's exit and the efficiency of the radial inlet chamber significantly changed with the variation of the grid number. However, when the number of grids increased to Mesh \(5=7.7\times 10^{6}\) elements, the flow and performance parameters almost remained unchanged, and the maximum deviations of the static pressures and efficiencies obtained by Mesh 5, Mesh 6 and Mesh 7 were only 0.006% and 0.019%, respectively. The results of the other flow parameters also indicate that, when the grid elements increased to the number marked with a red line in the figure, there was little difference in the numerical results obtained with different grid resolutions. Therefore, Mesh 5 was considered to ensure sufficient accuracy for the numerical simulation of this radial inlet model. For the first stage of the compressor downstream from the radial inlet chamber, the grid convergence study also proved that the probable uncertainties caused by the computational grids were less than 0.03% [31]. The final computational grids for different components of the CFD models are shown in Figure 4.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**No. ** & **Inlet Pipe** & **Radial Inlet Chamber** & **Guide Vanes** & **Outlet Pipe** & \begin{tabular}{c} **Total Grid** \\ **Number** \\ \end{tabular}  \\ \hline \(1\) & \(1.4\times 10^{5}\) & \(5.8\times 10^{5}\) & \(6.0\times 10^{5}\) & \(5.1\times 10^{5}\) & \(1.8\times 10^{6}\) \\ \(2\) & \(2.2\times 10^{5}\) & \(8.9\times 10^{5}\) & \(9.0\times 10^{5}\) & \(8.1\times 10^{5}\) & \(2.8\times 10^{6}\) \\ \Considering that different turbulence models have specific requirements for the distance from the first node of the grid to the solid wall, the \(y^{+}\) values obtained with the above computational grids were checked in Figure 5. It can be found that the \(y^{+}\) values of over 90% for the solid surfaces were in the range of 1-8, and the maximum \(y^{+}\) value was less than 10, which met the requirement of the Spalart-Allmaras turbulence model employed in this paper.