## 5 Conclusions  
In order to verify and compare the reliability of state-of-the-art correlations for the prediction of leakage mass flow through a labyrinth seal, the results of a wide experimental campaign performed on both a straight-through and a stepped seal are compared to the predictions estimated through the most-known analytical approaches. In general, in the evaluation of the results, attention was paid, not only to investigate the correlations that provided the closest match to the experimental findings, but especially to highlight the precision in the modeling of the impact of different parameters through the evaluation of the error trends. A dedicated test rig was used to characterize the seals behavior by varying the overall pressure ratio and the Reynolds number independently from each other; this allowed us to acquire test points by changing \(PR\) and \(Re\) simultaneously (by keeping the outlet pressure unaltered and varying the inlet one) and by changing PR with constant Reynolds (by varying both inlet and outlet pressure). In this way, it was possible to evaluate the accuracy of the considered correlative approaches and highlighting their limitation in modeling the impact of these two parameters on the resulting leakage mass flow rate. In general, the following indications could be provided:  
* For the straight-through seal and nominal clearance size, the impact of the Reynolds number was overestimated by the Zimmermann correlation, whereas a better modeling was achieved through Vermes and Hodkinson ones. * For the stepped seal, the need for improvement in the modeling of the impact of the overall pressure ratio was evidenced, for all approaches. Carry-over effects must be also neglected to achieve a proper prediction. * Concerning the evaluated pressure distribution across the seal, thanks to the indirect models, the possibility to distinguish between the behavior of the first and the following constrictions is mandatory to achieve a reliable prediction for the straight-through seal. * Concerning different clearance sizes, for the straight-through seal, Morrison correlation provided the best matching for the straight-through seal at the nominal clearance size, with errors within 5% in the tested \(Re\)-\(PR\) range; on the other hand, errors reached \(\pm\)20%, if all clearance values were considered. Vermes correlation, while being among the worst-performing for the nominal clearance size (errors around 10%), is the only one that can correctly predict the performance trend induced by the clearance variation. * Similar considerations could be made for the stepped seal, with more limited variations and overall errors. Figure 15: Comparison between experimental and analytical discharge coefficient for increasing values of the clearance-to-tip width ratio (stepped labyrinth—nominal \(PR\)): (a) predicted/measured values and (b) relative differences. * Overall, Vermes correlation has been revealed to be the best performing. It provides the more accurate modeling of different aspects, thus returning correct trends, despite a certain offset, with respect to experimental findings. * The adoption of a correlation able to correctly predict the behavior of the seal at different clearance sizes is crucial since its value is generally known with a limited accuracy during the engine operation and lifetime. The benchmark of a certain correlative approach for different values of clearance sizes is, therefore, evidenced as a mandatory step. Conceptualization, T.B., A.P., B.F. and L.W. ; methodology, T.B., A.P. and B.F.; software, N.C., T.B. and A.P. ; validation, N.C., T.B., A.P. and L.W. ; formal analysis, N.C., T.B. and A.P. ; investigation, N.C., T.B. and A.P. ; resources, N.C., T.B., A.P. and L.W. ; data curation, N.C. and T.B. ; writing--original draft preparation, N.C. and T.B. ; writing--review and editing, N.C. and T.B. ; supervision, B.F.; project administration, L.W. ; funding acquisition, B.F. and L.W. All authors have read and agreed to the published version of the manuscript.