## 4 Results
### Stepped Labyrinth Comparison Results
Since the same decreasing trend is achieved regardless of the Reynolds value and variation rate, it can be stated that the mispredicted behavior stays in the dependence on the overall pressure ratio, through the \(\beta\) term in Equation (1) or its influence on the flow coefficient, for the stepped seals. Figure 14 shows the pressure distribution across the stepped labyrinth for two investigated overall pressure ratios. As already shown for the straight-through labyrinth, at the lower \(PR\), the indirect methods are able to match the experimental data. At the higher \(PR\) value, the Neumann and Kurohashi series collapse due to the carry-over removal but they underestimate the real values from 3% to 12% at the second and seventh tooth, respectively. On the other hand, [PERSON] slightly overestimates the actual trend but the percentage differences are below 10% for all the pressures. It can be seen that the pressure step at the first tooth disappeared and a smoother curve is obtained. This is mainly due to the removal of the carry-over factor from Equation (6) and it can be ascribed to the effect of the step height \(s\) on the kinetic energy. Its almost complete conversion into pressure energy inside  
Figure 11: Comparison between experimental mass flow data and calculated values coming from the correlations—carry-over factor is included in the indirect methods (stepped labyrinth—nominal \(c\) value): (**a**) predicted/measured values and (**b**) relative differences  
Figure 12: Comparison between experimental mass flow data and calculated values coming from the correlations—carry-over factor is neglected in the indirect methods (stepped labyrinth—nominal \(c\) value): (**a**) predicted/measured values and (**b**) relative differences  
the seal cavity makes the flow field after each tooth more similar to that just before the first one. Eventually, the discharge coefficient is plotted as a function of the clearance-to-tip width ratio in Figure 15, with the same approach used for the straight seal. A slightly decreasing trend can be highlighted in Figure 15a, even if the trend is not as smooth as the one found for the straight-through labyrinth. Looking at the series coming from the correlations, both [PERSON] and [PERSON] are able to capture the decreasing trend while an almost constant value is predicted by the others. Since carry-over effect have been shown to be negligible for stepped seals, this decreasing trend is surely due to the impact of flow coefficient in Equations (2) and (6). As stated in [13], [PERSON] suggested a flow coefficient extracted from available data derived from dedicated tests performed on these kinds of labyrinth. On the other hand, a decreasing value of the flow coefficient is prescribed for increasing clearance values, according to [PERSON] [2]. Despite the not perfectly smooth experimental trend, [PERSON] correlation is the one that provides the best prediction of the overall \(C_{d}\) reduction rate, as the clearance size is increased. Figure 15b highlights that for all the undertaken methods the values ranges from \(-10\%\) to \(+10\%\). [PERSON] and Kurohashi models show, overall, the lowest relative differences, as their constant predicted value can be read as a mean value of the experimental discharge coefficients. To conclude, it is worth reminding that the proposed correlations do not include all the geometrical parameters of the seals. For instance, the height of the step \(s\) is supposed to be the key in promoting the dissipation of kinetic energy and determining the flow path inside the cavity. In addition, validity ranges are not fulfilled for the Morrison correlation both for the straight-through and the stepped seal, since the \(w/c\) ratio exceeds the prescribed values. Figure 14: Pressure distribution across the stepped labyrinth for two values (\(PR=1.5\) and \(PR=5.0\)) of the overall pressure ratio (nominal \(c\) value).