## 3 Experimental Facility  
The flow check tests were performed on both a straight-through and a stepped seal thanks to a dedicated experimental apparatus, developed in collaboration with Baker Hughes--Nuovo Pignone Tecnologie s.r.l. and installed in the Heat Transfer and Combustion Laboratory of the Department of Industrial Engineering of the University of Florence (DIEF). A non-rotating test rig was designed, thus neglecting all the effects related to the rotation. These simplifications were justified by [PERSON] et al. [22] and by [PERSON] et al. [8], who showed that, when axial velocity component of flow above the tooth's tip is higher than the circumferential velocity of rotor tip, the effect of rotation hardly affect the results. This condition was verified for the engine conditions that the present tests were aimed to simulate. Moreover, tests were carried out on linear test articles (i.e., representing a linearized slice of the full annular engine configuration), with a width (\(W\)) equal to around 200 times the clearance dimensions, thus minimizing border effects. The experimental campaign was carried out in scaled operating conditions, in order to improve the dimensional \begin{table}
\begin{tabular}{l l l l} \hline \hline  & **Flow Coefficient, \(K\)** & **Carry-Over Factor, \(\mu\)** & **Other** \\ \hline Direct Methods & & & \\ [PERSON] [11] & Constant value [10] & \(f(N,c/t)\) & - \\ [PERSON] [2] & \(f(Re,w/c)\)[12] & \(f(t,w,c)\) & - \\ [PERSON] [13] & \(f(Re,PR,w/c)\) & \(f(N,c/t)\) & - \\ \hline Indirect Methods & & & \\ [PERSON] [16] & Chaplygin formula, & & \\  & \(f(\beta_{i},\gamma)\)[17] & & \(f(N,c/t)\) & - \\ [PERSON] [20] & Chaplygin formula, & & \\  & \(f(\beta_{i},\gamma)\)[17] & & \(f(N_{i},K,c/t)\) & - \\ [PERSON] [21] & - & - & Discharge coefficient \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Correlation summary. accuracy, by increasing the test samples dimensions (a scale factor close to 3 was achieved), and to allow for an easier experimental approach with more manageable conditions. The test samples are representative of real labyrinth seals installed on BH engines. Each test sample is composed by four parts: the test article with the teeth, two lateral components, defining the resulting clearance size, and a top cover. All the components are made with stainless steel, by CNC or EDM (for the components with teeth) machining, and carefully assembled in order to reproduce the proper flow path and the right dimension of the clearance gap, as shown in Figure 2(a),b. The test rig is composed of a single fluid line (Figure 4). A compressor able of delivering air at a maximum pressure of 10 bar feeds the test rig. The air coming from the compressor is first dried, in order to prevent condensation during the expansion inside the test article. A pressure regulator provides a constant line-inlet pressure, while the mass flow rate (i.e., Reynolds number) through the rig is regulated by means of a chocked manual valve, upstream of the rig. Downstream of the second valve, a calibrated orifice flow meter is installed (\(\pm\)1.0% accuracy). A separation plate divides the inlet plenum (i.e., test article feeding) and the discharge plenum. Since a uniform feeding of the test sample is required, the airflow enters the upper part of the inlet plenum radially, and then, a perforated plate is installed to damp the velocity components in non-axial direction. The test sample is located in the second chamber of the inlet plenum, installed on the separation plate. In order to provide a better understanding of the whole installation, a sketch of the test rig is provided in Figure 5. The discharge pressure can reach below-ambient conditions thanks to vacuum pumps, in order to maximize the achievable pressure ratio (\(PR\)) across the seal. This is regulated by a downstream manual valve and/or by the speed of the vacuum pumps. The upstream valve remains chocked throughout the whole operation, thus allowing to decouple Reynolds and \(PR\) regulation and to easily set the operating conditions.