## 3 Analysis and Discussion of the Results
### Validation of the CFD Simulation and Analysis of the Loss Behaviour
The value of \(\overline{w}\) decreases from \(s=0.15\) for \(\Delta\alpha_{\text{pb}}=0^{\circ}\) until the impeller outlet. A minimum is observed at \(s=0.8\). The specific impeller work described in [27] can also be expressed as follows:  
\[\widetilde{Y}=\frac{1}{2}\Big{(}u_{2}^{2}-u_{\text{I,m}}^{2}+w_{\text{I,m}}^{ 2}-w_{2}^{2}+c_{2}^{2}-c_{\text{I,m}}^{2}\Big{)}. \tag{14}\]  
This shows that the difference in relative velocity must be larger for negative \(\Delta\alpha_{\text{pb}}\). With positive \(\Delta\alpha_{\text{pb}}\), on the other hand, there is a smaller difference. In addition to the whirl and the relative velocity, the mass flow averaged entropy production (\(\overline{S_{\text{pro}}}\)) over the dimensionless streamline length was also analyzed in Figure 11c. The entropy production exhibits a pronounced increase at \(s=0.15\) as it enters the impeller blade leading edge. The largest increase is observed at \(\Delta\alpha_{\text{pb}}=-30^{\circ}\). The increase in entropy production is then almost linear for \(\Delta\alpha_{\text{pb}}\in\{0^{\circ};+30^{\circ}\}\) in the blade channel up to \(s=0.9\). In accordance with Figure 10b, the losses for \(\Delta\alpha_{\text{pb}}=-30^{\circ}\) are higher in the entire blade channel compared to the other two curves. The relative velocity exerts a determining influence on the Coriolis acceleration (\(a_{\text{c}}\)), as demonstrated by Equation (15) [27]. The symbol \(\underline{w}\) denotes the vector of relative velocity, while the symbol \(\underline{w}\) denotes the vector of angular velocity. The Coriolis force is counteracted by the centrifugal forces due to the streamline curvature and the rotation of the impeller [27]. Should the centrifugal or Coriolis forces increase significantly in comparison to the other force, secondary flows and associated turbulence losses will occur in the impeller. The modified Rossby number, as defined by Equation (16) [33], can be employed to assess the secondary flows. \[Ro_{\text{m}}=\frac{6\omega r}{2w} \tag{16}\]For \(Ro_{\text{m}}>1\), the centrifugal forces prevail, resulting in a deflection of the flow towards the suction side of the blade. Conversely, for \(Ro_{\text{m}}<1\), the flow is driven away from the suction side. In Figure 11d, the area-averaged modified Rossby number (\(\overline{Ro_{\text{m}}}\)) is plotted against the dimensionless streamline length (\(s\)) for all three prewhirl body angles. For all three whirl angles, the modified Rossby number in the range \(0.4<s<0.6\) is in the range of one, so that reduced secondary flows can be expected here. A comparison of the modified Rossby number curve with that of entropy production in Figure 11c reveals that outside the interval \(Ro_{\text{m}}\in\{0.6;1.4\}\), there are significant flow losses. In their study, [PERSON] et al. posit that the observed increase in entropy production at the blade trailing edge can be attributed to dynamic static interference [23]. Figure 11d demonstrates that in the vicinity of the blade exit edge, the modified Rossby numbers undergo a decrease, which can be attributed to an increase in the Coriolis force. This results in the flow deflection observed by [PERSON] and [PERSON], which in turn leads to a reduction in performance [27; 34]. Furthermore, an exchange of momentum occurs as a result of augmented secondary flows. The rising Coriolis force can be attributed to the accelerating relative velocity, as illustrated in Figure 11b. The rise in entropy production is therefore evidently justified and can be characterised by the relative velocity. As stated in [27], the specific losses in the impeller (\(Z_{\text{LN}}\)) can be subdivided into two categories: shock losses in the impeller inlet (\(Z_{\text{LN,St}}\)) and friction and turbulence losses in the blade channel (\(Z_{\text{LN,Ka}}\)):  
\[Z_{\text{LN}}=Z_{\text{LN,St}}+Z_{\text{LN,Ka}}. \tag{17}\]  
The relationship \(Z_{\text{LN,St}}\propto{w_{1}}^{2}\) is specified for the shock losses in [27].