## 2 Computational Model
### Mesh Generation  
Two detached meshes were generated, one each for the impeller domain and the casing domain using different topologies, as shown in Figure 2. Structured mesh for the impeller's domain using hexahedral elements was constructed. While unstructured mesh for the casing was generated using tetrahedral and prism elements, prism elements were only used in the prism layers near the wall for the flow in the boundary layer. To accomplish the requirements of the SST turbulence model, the value of y\({}^{+}\) was guaranteed to be less than one for both stationary and rotating domains to acquire full advantage from the abilities of the SST turbulence model [19]. The y\({}^{+}\) is a function of \(\Delta y\) (first node's distance from the wall), flow length scale (\(L\)) and flow Reynolds number (Re). The value of \(\Delta y\) was calculated from Equation (12) to accomplish the desired value of y\({}^{+}\)[37]. Further details on these calculations can be found in [30; 38]. Mesh details are provided in Table 2.  
\[\Delta y=L\ y^{+}\sqrt{74}\ Re_{L}^{-\frac{13}{14}} \tag{12}\]  
Figure 2: Mesh for rotating and stationary domains.  
For accurate calculations, attaining only a desired \(\text{y}^{+}\) value is not sufficient. Additionally, a minimum number of nodes (\(n_{min}\)) within the boundary layer thickness (\(\delta\)) are also required. For the SST turbulence model, the condition for (\(n_{min}\)) is exemplified in Equation (13) [37].  
\[\left\{\begin{array}{rl}y_{n}(n_{min})\ \leq\delta\\ for\ SST\ n_{min}=15\\ y_{n}(15)\ \leq\delta\end{array}\right. \tag{13}\]  
where \(y_{n}(15)\) is the distance of node 15 from the wall.  
The value of \(\delta\) corresponding to \(Re\) and \(L\) was calculated from the following equation:  
\[\delta=0.035\ LRe_{L}^{-\frac{1}{2}} \tag{14}\]  
The computed value of the boundary layer thickness was taken as equivalent to \(y_{n}(n_{min})\) and the mesh growth rate (\(r_{m}\)) from the wall side was computed employing the geometric progression relation provided in Equation (16) to accomplish the desired value of the \(n_{min}\) within the boundary layer thickness.  
\[r_{m}=\left[\frac{y_{N}(n_{min})}{\Delta y}\right]^{\frac{1}{n_{min}-1}} \tag{15}\]  
\[r_{m}=\left[\frac{y_{n}(15)}{\Delta y}\right]^{1/14} \tag{16}\]  
It is to be noted here that Equations (12) and (14) 4 were derived for a flat plate, but authors could not find any other method in the literature to adjust the value of y+. Therefore, in this work, Equations (12)-(14) were used to get an initial estimate of \(\Delta y\). Later an iterative process was used to achieve the required value of y+. The iterative process involved the following steps: (1) estimation of \(\Delta y\) from the equation node distribution in the boundary layer region; (2) updating the mesh based on the calculation in step; (3) solving the mesh in CFX; (4) postprocessing y+ from the obtained solution and finally adjusting the \(\Delta y\) values accordingly. The process mentioned above takes three to four  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Rotating Domain (Impeller)**} & \multicolumn{3}{c}{**Stationary Domain**} \\  & & & & & & **(Casing)** & & & \\ \hline M1 & 35 & 10 & 5 & 15 & 0.003 & 1,025,341 & 11,325 & 57 & 11.2 \\ M2 & 45 & 20 & 7 & 20 & 0.002 & 1,523,452 & 18,256 & 97 & 10.7 \\ M3 & 60 & 30 & 11 & 25 & 0.0015 & 2,742,863 & 35,261 & 195 & 9.26 \\ M4 & 70 & 40 & 15 & 30 & 0.001 & 3,929,536 & 47,325 & 310 & 9.19 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Mesh details for the mesh optimization study.  
iterations to achieve the required values of y+. Further, the chord length of the impellor's blade was taken as the characteristic length \"L\" for these calculations