## 3 Analysis of the Impeller-Diffuser Interaction  
### Steady Analysis at 2M  
The time-averaged quantities were compared to the corresponding values for the steady analysis and the experimental data. Figures 6 and 7 show the pitchwise and spanwise distributions of \(\mathrm{p_{t}}\) for the operating points M and P1. The numerical and experimental data show the same trend, with a tendency of the numerical results to underestimate the total pressure values; the difference is comparatively small and reaches 10.9% at its maximum. In the case of P1 point (Figure 7), the total pressure is well-captured using the k-\(\omega\) turbulence model for up to 50% of the span. The model results become less accurate moving near the diffuser front wall. In this zone (z/b = 0.7 and z/b = 0.9), characterized by a strong whirling flow, the SST simulations seem to provide a better matching with the test case reference values. The spanwise distribution of \(\mathrm{p_{t}}\) has a maximum at z/b = 0.7 for both the operating points M and P1, while the circumferential distribution is almost constant: potential flow effects do not influence the total pressure distribution, which is only determined by the impeller geometry and operating point (inlet conditions, rotational speed). The discussion of the z/b distributions needs a preliminary investigation on the contributions of static (p) and dynamic (dependent on Ma) components that make  
Figure 4: Mach number distribution at z/b = 0.5 for steady simulations HR k-\(\omega\) (left) and HR-SST (right) in the operating point P1.  
Figure 5: Total pressure ratio at plane 8M.  
up the total pressure values. Only the simulation HR-KO will be considered for the point P1, because of its higher numerical stability and better coherence with the experimental data.  
The maximum of \(\mathrm{p_{t}}\) at z/b = 0.7 is merely caused by the dynamic component, as shown by the corresponding maximum of Ma in Figure 8, because the static pressure is almost constant along z/b in 2M (z/b distribution of p in Figure 8). Because of the low meridional velocity at the surge limit, the flow is essentially tangential in 2M. Therefore, it attacks with a high incidence angle the stator blade leading edge: the stator potential flow field causes a much greater blockage (i.e., higher pressure and lower Mach number) near the diffuser PS than near the SS, as can be seen in the pitchwise distributions of p and Ma (Figure 8).  
Assuming the inviscid flow hypothesis, the velocity variations in the meridional and blade-to-blade planes can be calculated separately, as first proposed by [PERSON] (1952) [25]. Neglecting the force due to the blade lean angle (\(\frac{\partial\beta_{\mathrm{blade}}}{\partial t}=0\)), the hub to shroud velocity variation in the meridional plane is given by the following relation, which has been applied at the impeller exit (n = z, \(R_{n}\)\(\rightarrow\)\(\infty\), \(\gamma\) = 90\({}^{\circ}\), u = constant):  
\[\frac{\partial w}{\partial n}=\frac{w\cdot\sin^{2}\beta}{R_{n}}-\cos\gamma \cdot\cos\beta\left(\frac{w\cdot\cos\beta}{r}-2\Omega\right)\Rightarrow\frac {\partial w}{\partial z}=\frac{\partial c}{\partial z}=0 \tag{1}\]  
Figure 6: Total pressure at 2M, point M.  
Figure 7: Total pressure at 2M section, point P1P1.  
where \(n\) is the coordinate perpendicular to the axisymmetric streamwise, \(R_{n}\) is the curvature radius of the meridional streamline and \(\gamma\) is the angle between streamline and axial direction in the meridional plane.  
Equation (1) suggests that, under the inviscid flow hypothesis, the hub to shroud velocity distribution is constant at the impeller exit. Therefore, the variable distribution of Ma along z/b shown in Figure 8 is due to viscous phenomena inside the impeller.