## 2 Materials and Methods
### CFD Simulation
#### 2.4.5 Discrete Model and Mesh Validation  
A structured mesh was generated within the stationary flow domain and within the rotational domain of the rotor, as shown in Figure 5a,b. Irregular triangular finite elements were generated in the rotational blade domains, as presented in Figure 6a. The boundary layer mesh was generated at the blade's walls using hexahedra elements, as shown in Figure 6b. The parameters of the inflation layer were adopted to provide the non-dimensional parameter \(y+<\)1 for each blade angular position during rotor operation. At this value of \(y+\), the boundary mesh is fine enough to directly resolve the viscous sublayer at the blade surface. Figure 4: Boundary conditions and dimensions of the computational domain. Figure 5: Mesh (**a**) flow domain, (**b**) rotor domain. It is obtained when the first cell on the wall is inside a viscous sublayer. The enhanced wall treatment in Ansys Fluent was obtained for \(y+\leq\)1. However, higher values up to \(y+<5\) are also allowed if the first cell is still within the viscous sublayer [29].The boundary layer mesh consisted of 30 layers, with the first layer height of 0.01 mm and a growth ratio of 1.15 [30]. The torque coefficient \(C_{m}\) and the power coefficient \(C_{p}\), given by Equations (19) and (20), were considered for mesh independence study. \[C_{m}\ =\ \frac{2M}{\rho W^{2}AR} \tag{19}\]  
\[C_{p}\ =\ C_{m}\lambda\ =\ C_{m}\frac{\omega_{R}R}{W} \tag{20}\]  
where \(A\)--the swept area of the rotor (\(A=D\cdot H\)), \(H\)--blade height. The mesh independence study was conducted considering the fourth rotation of the rotor with one dual-coherent blade. The mesh quality was refined by changing element sizes in all the domains. However, the quality of the boundary layer mesh was very fine in all cases to provide \(y+<\)1. The results of mesh validation referring to the average values of \(C_{m}\) and \(C_{p}\) are summarized in Table 2.  
It can be noticed that average values of the torque coefficient and the power coefficient were converged for medium and fine mesh. Therefore, the mesh of medium quality was used for all simulations. Figure 7 shows the variation in \(C_{m}\), depending on the rotor angle \(\alpha\), for the fourth rotation. It can be observed that there were only slight differences in \(C_{m}\) values in the angle ranges of 140\({}^{\circ}\)-170\({}^{\circ}\), 260\({}^{\circ}\)-270\({}^{\circ}\), and 290\({}^{\circ}\)-295\({}^{\circ}\), which were negligibly small and did not affect the average \(C_{m}\) value for the whole rotation. \begin{table}
\begin{tabular}{c c c c} \hline
**Mesh Quality** & **Number of Elements** & \(Cm\) & \(Cp\) \\ \hline Coarse & 10,061 & 0.225 & 0.061 \\ Medium-coarse & 24,895 & 0.256 & 0.256 \\ Medium & 71,220 & 0.260 & 0.070 \\ Fine & 285,382 & 0.261 & 0.071 \\ \hline \end{tabular}
\end{table} 
Table 2: Mesh validation. Figure 6: Mesh (a) blade domain, (b) boundary layer. The quality of the mesh was also evaluated in the regime of two main variables capturing the quality of the numerical flow solution. The first variable was the dimensionless wall distance \(y+\). It is commonly accepted that the value of \(y+<1\)[39] is required to obtain enhanced wall treatment and to directly resolve the viscous sublayer. The second variable was the dimensionless Courant number obtained using the Courant-Friedrichs-Lewy (CFL) criterion given by Equation (21). The CFL criterion is a convergence condition that incorporates the ratio between the time step and a cell size. In general, when using explicit integration schemes, it is recommended to obtain \(Co<1\)[40]. However, concerning viscous turbomachinery flows, solving using an implicit scheme is recommended to obtain larger values of the Courant number, about \(Co<10\)[32] or \(Co<20\)[40]. \[C\ =\ \overrightarrow{v}\frac{dt}{dx} \tag{21}\]  
where \(\overrightarrow{v}\)--the velocity vector, \(dt\)--the time step, \(dx\)--the cell size.