## 4 Results  
Before presenting the main outcomes of the above-mentioned correlative approaches, some results of the experimental campaign are shown. Tables 4 and 5 shows the complete test matrix of the second regulation strategy and the corresponding discharge coefficient for each test point; for a given pressure ratio value, the percentage difference is provided for each Reynolds number compared to the lowest one. \begin{table}
\begin{tabular}{l l l l l l l} \hline  & **0.6\(Re_{nom}\)** & **0.75\(Re_{nom}\)** & **0.9\(Re_{nom}\)** & **1.1\(Re_{nom}\)** & **1.25\(Re_{nom}\)** & **1.4\(Re_{nom}\)** \\ \hline \(PR=1.2\) & 0.409 (1) & 0.414 & 0.421 & 0.424 & 0.426 & 0.427 \\  & & (+1.17\%) & (+2.91\%) & (+3.72\%) & (+4.33\%) & (+4.33\%) \\ \(PR=1.5\) & 0.433 (1) & 0.440 & 0.448 & 0.452 & 0.453 & 0.452 \\  & & (+1.66\%) & (+3.57\%) & (+4.40\%) & (+4.73\%) & (+4.40\%) \\ \(PR_{nom}\) & 0.443 (1) & 0.454 & 0.461 & 0.466 & 0.467 & 0.467 \\  & & (+2.40\%) & (+4.02\%) & (+4.99\%) & (+5.39\%) & (+5.29\%) \\ \(PR=2.0\) & 0.466 (1) & 0.476 & 0.485 & 0.490 & 0.492 & 0.493 \\  & & (+2.11\%) & (+4.11\%) & (+5.23\%) & (+5.64\%) & (+5.81\%) \\ \(PR=2.25\) & 0.476 (1) & 0.487 & 0.497 & 0.502 & 0.503 & 0.505 \\  & & (+2.18\%) & (+4.29\%) & (+5.43\%) & (+5.68\%) & (+6.11\%) \\ \(PR=2.5\) & 0.483 (1) & 0.492 & 0.504 & 0.509 & 0.512 & 0.513 \\  & & (+1.97\%) & (+4.27\%) & (+5.42\%) & (+6.01\%) & (+6.25\%) \\ \(PR=3.0\) & 0.491 (1) & 0.500 & 0.510 & 0.515 & 0.518 & 0.519 \\  & & (+1.81\%) & (+3.97\%) & (+4.89\%) & (+5.54\%) & (+5.72\%) \\  & & 0.505 & 0.512 & 0.518 & 0.520 & 0.521 \\  & & (+2.05\%) & (+3.56\%) & (+4.69\%) & (+5.13\%) & (+5.32\%) \\ \(PR=4.0\) & 0.496 (1) & 0.505 & 0.516 & 0.519 & 0.521 & 0.523 \\  & & (+1.83\%) & (+4.09\%) & (+4.79\%) & (+5.13\%) & (+5.48\%) \\ \hline \end{tabular}
\end{table} 
Table 4: Experimental discharge coefficient for each test Reynolds number and pressure ratio (straight-through labyrinth). \begin{table}
\begin{tabular}{l l l l l l l} \hline  & **N** & **\(c\)/w** & **\(c\)/t** & **c/h** & **s/c** & **\(\alpha\)** & \(\beta\)** \\ \hline Straight-through seal & 5 & 0.36–0.84 & 0.017–0.039 & 0.031–0.072 & 0.00 & 31.5\({}^{\circ}\) & 15\({}^{\circ}\) \\ Stepped seal & 7 & 1.12–2.44 & 0.029–0.063 & 0.042–0.091 & 5.46–11.8 & 37.5\({}^{\circ}\) & 15\({}^{\circ}\) \\ \hline \end{tabular}
\end{table} 
Table 3: Test sample geometrical features. The data of Tables 4 and 5 are plotted in Figure 5(a) and Figure 5(b), respectively. They show the discharge coefficient as a function of the overall pressure ratio for different values of the tested Reynolds numbers for the straight-through and the stepped labyrinth seals, respectively, when the nominal value of the tip clearance gap is considered. It can be noticed that the higher the pressure ratio, the higher the \(C_{d}\). However, as the pressure ratio increases, a well-established asymptotic trend is achieved. In addition, the Reynolds number has a very poor effect on the trends since the discharge coefficient slightly increases with the Reynolds number for a given value of pressure ratio. As shown by Figure 6 and by the percentage differences in the previous Tables, the values of the stepped seal overlap almost perfectly; for the straight-through seal, a 5% difference can be highlighted between the highest and the lowest Reynolds, hinting that, for this geometry, the asymptotic behavior is reached at a slightly higher Reynolds number. It can be speculated, even if not demonstrated by the present experimental research, that this is due to the higher momentum of the jet exiting from the constriction, resulting in higher carry-over effects (which are not present for the stepped seal). As a result, for both the tested labyrinth seals, the dependency of the discharge coefficient on the Reynolds number can be neglected and, in the following, most of the results will be presented at different flow Reynolds numbers without losing generality.