## 4 Results
### Straight-Through Labyrinth Comparison Results  
In this section, the main results coming from the correlative models described in Section 2 are presented for the straight-through seal. Firstly, the estimated mass flow rate is compared with the measured one for the nominal value of the clearance. The estimated pressure distribution along the seal is shown for all the indirect models. Then, since one of the main contributions of this paper is to evaluate the relationship between the discharge coefficient and the clearance value, this trend is investigated for each correlation. The experimental data shown in Figure 7a refer to the first regulation strategy explained in Section 3, hence both the Reynolds number and the overall pressure ratio change at each test point, as the discharge pressure is kept constant. Figure 7a shows the leakage mass flow rate as a function of the overall pressure ratio for the nominal clearance value. The expected increasing trend can be highlighted. The black solid line represents the experimental measurements, while the dashed lines depict the correlative methods. Both the direct and the indirect models are obviously able to reproduce the increasing trend and to provide an acceptable data matching, since the differences with respect to the experiments are, in general, quite limited. A deeper insight is provided by Figure 7b. It shows the percentage difference between the measured and estimated mass flow rate for each overall pressure ratio. All the correlation, except the Morrison one, underestimate the actual value of mass flow. In addition, all the indirect methods highlight an almost constant relative difference with varying PR. The direct methods, on the other hand, do not show the same behavior; [PERSON] and Hodkinson correlations provide a decreasing trend (i.e., increasing underestimation) for low PR, before settling to approximately constant values. The Zimmermann equation shows an increasing trend, as the underestimation reduces for increasing values of \(PR\), and reaches an almost perfect matching (null relative difference with experiments) at the maximum tested PR. This is mainly due to the way the flow coefficient is modeled in those equations, since the same \(\beta\) term is included and the carry-over factor is fixed for a given clearance value, depending only on geometrical features. As stated in Section 2.1.1, [PERSON] suggested the employment of a constant K value, while the one provided by [PERSON] slightly depends on the Reynolds number (i.e., mass flow rate) and only a 4% increase is achieved when the Reynolds number is increased from the lowest to the highest value. A completely different situation shows up if the Zimmermann model is considered since the flow coefficient variation between the lowest and the highest Reynolds number is about 16%: the present benchmark shows that such a variation rate overestimates the actual one, at least for the tested seal geometry. In order to provide a deeper comprehension of this dependence, Figure 8 shows the leakage mass flow percentage difference as a function of the overall pressure ratio when test points coming from the second regulation strategy (i.e., varying PR with constant Reynolds) are considered. The plot is referred to a flow Reynolds number of 5000 but similar results are found, and hence the same comments can be pointed out, if another \(Re\) number is taken. Although a slightly decreasing trend is achieved at the lowest \(PR\) values, a constant pattern is then achieved at higher pressure ratios for all the direct methods discussed so far: since a constant Reynolds results in a constant value of flow coefficient predicted by the considered direct methods, it is confirmed that the error variation evidenced in Figure 7b for Zimmermann approach is due to an incorrect modeling of the K-Re dependency, for the investigated geometry.