## 4 Results
### Stepped Labyrinth Comparison Results  
In this section, the main results coming from the correlative analysis are presented for the stepped labyrinth seal. As already performed for the straight-through one, the estimated mass flow rate is compared with the measured one for the nominal value of the clearance. However, two different and complementary approaches are shown. Then, the pressure distribution coming from the indirect models is presented. Finally, the discharge coefficient as a function of the clearance size is discussed. As already explained, the proposed analytical models were developed during the past decades specifically for straight-through labyrinths starting from the available empirical data. However, many authors recommend their application also for stepped seals as long as certain assumptions were made. In particular, [PERSON] [10] and [PERSON] [2] stated that the carry-over coefficient should be neglected for stepped labyrinths. By making the step \(s\) sufficiently high, all the kinetic energy of the jet will be dissipated before the flow enters the next cavity. Hence, it was decided to eliminate the \(\mu\) parameter from Equation (2). On the other hand, no indications are given for the indirect models. Thus, two different approaches have been employed; in the first case, the contribution of the carry-over factor \(\mu_{i}\) was kept inside Equation (6) and, in the second case, it was removed. Results are shown in Figures 11 and 12. In particular, Figures 11a and 12a show that the mass flow rate increases for increasing values of the overall pressure ratio for the nominal clearance value. The black solid lines represent the experimental measurements, while the dashed lines depict the correlative methods. It is clear from Figure 11a that neglecting the carry-over factor in the direct models allows a proper experimental data fitting, whereas its employment inside Equation (6) determines up to a 60% overestimation of the actual mass flow (see Figure 11b) since the dissipation of kinetic energy is not correctly modeled. However, as shown by Figure 12a, a proper trend is restored whether the carry-over factor is neglected also for the indirect models. The percentage differences fall down to 16% (Figure 12b) in the worst case scenario represented by the Morrison correlation; it should be noted that \(w/c\) validity range is not fulfilled for the Morrison correlation and hence, the model could lose reliability. The correlative approaches mostly tend to overestimate the leakage mass flow, in a slightly decreasing extent, as the PR is increased; an increasing trend occurs only for the methods from [PERSON] and [PERSON]. After \(\mu_{i}\) is removed from Equation (6), there is no difference between [PERSON] and [PERSON] models since they differs only by the carry-over factor formulation. Hence, both correlations gave the same results; this is clearly visible in Figure 12b, as [PERSON] and Kurohashi series perfectly overlap. Figure 10: Comparison between experimental and analytical discharge coefficient for increasing values of the clearance-to-tip width ratio (straight-through labyrinth—nominal _PR_): (**a**) predicted/measured values and (**b**) relative differences. Looking at the direct methods proposed in the present work, the percentage differences between experiments and correlations are plotted in Figures 12b and 13, for first and second regulation strategy, respectively. In both cases, all the analytical trends are monotonically decreasing as the \(PR\) increases. In particular, Hodkinson and Vermes lines have the same shape as, they both suggest a constant \(K\) value in the range of tested Reynolds number and clearance size. Further, [PERSON] does not provide a constant trend for a fixed Reynolds number even if he models the flow coefficient as a function of the \(PR\)[14].