## 4 Results
### Straight-Through Labyrinth Comparison Results
As stated above, the indirect models allow to retrieve the pressure distribution along the labyrinth, i.e., the pressure values downstream each tooth, knowing the inlet and the discharge pressures. Figure 9 shows the ratio \(p_{n}/p_{N}\) for two tested overall pressure ratios. It can be highlighted that, at the lower \(PR\), the correlations can predict the experimental data with an almost-perfect fitting, while at the higher \(PR\) values, as the pressure variations are increased and the modeling is made more challenging, some differences arise, albeit quite limited ones. In particular, it should be noted that the steep pressure drop at the first tooth is not correctly reproduced by the Neumann model, since it does not distinguish between the different contribution of the first tooth and the following ones. On the other hand, bothKurohashi and [PERSON] developed a dedicated formula to accurately characterize the influence of the flow field approaching the first tooth and the one developing inside each cavity and approaching the subsequent teeth. As a result, the pressure drop across the first tooth is properly reproduced, while an underestimation of the following teeth pressure is achieved. Finally, the discharge coefficient is plotted for the nominal value of the pressure ratio as a function of the clearance-to-tip width ratio. Since the actual clearance value can change during the labyrinth life-cycle due to wear or thermal effects, it is key to evaluate the performance of the labyrinth for different value of the clearance gap size. Figure 10a shows the calculated discharge coefficients, while in Figure 10b, correlative results are scaled with experimental ones to produce relative differences. The black solid line in Figure 10a represents the experimental data: a decreasing trend can be highlighted. This means that, when increasing the clearance size, a reduction in carry-over factor or flow coefficient is achieved (or of both of them), but the precise reason can not be discerned from the available experimental results. The former (i.e., decrease in carry-over) can be achieved depending on the configuration of the jet exiting from one constriction; the latter (i.e., decrease in flow coefficient) can be generated by a limited variation of the actual area value \(A_{j}\) (see Figure 1), due to the flow field configuration inside the cavity and the resulting development of the vena contracta at the tooth tip: as the discharge coefficient values are calculated using the geometrical area \(A\), which scales linearly with the clearance size, the resulting \(C_{d}\) trend would be decreasing. It must be pointed out that other studies [5] reported opposite results,  
Figure 8: Percentage difference between experimental and analytical mass flow rates for tests at fixed Reynolds number value (straight-through labyrinth—\(Re=5000\)—nominal \(c\) value). Figure 7: Comparison between experimental mass flow data and calculated values coming from the correlations (straight-through labyrinth—nominal \(c\) value): (**a**) predicted/measured values and (**b**) relative differences  
with increasing \(C_{d}\) trends as the clearance gap increases. As the main difference between those and the present study stays in the teeth inclination (straight for the referenced works and inclined backward for the present one), this is thought to be the main parameter determining the evidenced trend. Looking back to Figure 10, only the Vermes correlation is able to reproduce the decreasing trend. It is worth reminding that all correlations return increasing carry-over factors, but [PERSON] is the only author that suggests the employment of a decreasing flow coefficient for increasing values of the clearance size, as reported in [2]. All the other models adopts an increasing flow coefficient for increasing clearance values and hence, they fail the prediction of the experimental trend. Since [PERSON] proposes increasing flow coefficient values for even higher gap sizes, this behavior is expected to reverse at some point (beyond tested values).