## 4 Results
### Inter blade Row Flow Field for Different Expansion Ratios: Op1-Op4  
In this section, the flow field in-between the blade rows is discussed for the four operating conditions, to provide an overview on the aerodynamics of the stator cascade and a reference for the subsequent discussion of the rotor-exit flow field. An overview of the stator-exit flow in transonic conditions (OP1) is shown in Figure 3 in terms of distributions of total pressure loss coefficient, Mach number, static pressure and stream-wise vorticity. The distributions of loss coefficient and Mach number show highly non-uniform flow in both pitch-wise and span-wise directions. The vane wake can be clearly identified in the loss distribution, enlarged by concentrated loss cores at 75% span and at 15% span. These concentrated loss cores roughly correspond to the secondary vortices, whose magnitude and orientation are marked by the stream-wise vorticity distribution. Two vortices appear in the tip region, namely the passage vortex (core I, negative) and the associated trailing edge shed vorticity (core II, positive). In the hub region, a weak passage vortex (core IV, positive) is found, while the trailing shed vorticity is smeared along the wake; at the bottom of the map the strongest hub vortex is found (core III, negative), which can be identified as a strong hub corner vortex. The Mach number and static pressure distribution also show large pitch-wise gradients, because of the vane potential field propagating downstream, and large span-wise gradients as a result of radial equilibrium. The combination between the two pressure fields leads to highest expansion ratio in the hub region (as visible in the static pressure map) inducing transonic flows at the bottom of the measurement area. At the left edge of this low pressure region, high pressure gradients are found, which are probably the trace of a weak shock wave generated on the rear suction side of the blade. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**OP** & \(G_{ND}\) & \(M_{n}\) & \(\beta_{\text{TS}}\) & \(\varphi\) & \(\psi\) \\ \hline OP4 & 0.205 & 0.218 & 1.155 & 5.00 & 2.09 \\ OP3 & 0.269 & 0.356 & 1.4 & 4.02 & 1.80 \\ OP2 & 0.299 & 0.458 & 1.65 & 3.47 & 1.58 \\ OP1 & 0.313 & 0.565 & 1.95 & 2.95 & 1.36 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Non-dimensional flow quantities for OP1-4 (reference diameter = 350 mm). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**OP** & **Flow Rate (kg/s)** & _Rpm_ & \(\beta_{\text{TS}}\) & \(T_{\text{Lin}}\) **0K** \\ \hline OP4 & 2.454 & 4150 & 1.155 & 303 \\ OP3 & 3.78 & 7000 & 1.40 & 323 \\ OP2 & 4.9 & 9000 & 1.65 & 323 \\ OP1 & 6.05 & 11,100 & 1.95 & 323 \\ OP3L & 3.75 & 5800 & 1.385 & 303 \\ OP3R & 3.75 By operating the turbine at the lowest expansion ratio (OP4), a similar flow pattern appears, with some relevant differences (Figure 4). Alongside the relevant reduction of peak Mach number, which goes down to \(\sim\)0.4, the wake appears significantly wider than that for OP1. This can be explained by resorting to the classic Mach and Reynolds number effects on an accelerating boundary layer where a higher acceleration tends to reduce the boundary layer thickness, and hence the thickness of the wake in transonic condition. The Mach number affects significantly the secondary flows, too; in the tip region for OP4, the vorticity distribution shows vortices with half of the magnitude with respect to that found at OP1; correspondingly, a wider secondary loss core is found for OP4 that evidences more smeared secondary flows in OP4 on the measurement traverse, thus suggesting a faster mixing process at low Mach number. In the hub region, the shed vorticity in the wake and the passage vortex (IV) can still be easily identified, though being very weak; the corner vortex, instead, is not even visible on the measurement plane. All these features are consistent with a reduction of the cross passage pressure gradient connected with a reduction in the expansion ratio.