## 1 Introduction  
One of the most important issues in turbomachinery design is the proper description, prediction and control of secondary flows. The prime example of this aspect is leakage flows. Efficient sealing systems must be employed to mitigate the total amount of air passing through the rotating and stationary components of the engine, since flow instabilities and, in general, efficiency reduction can occur if clearances and mass flow rates are too large. On the other hand, very small gaps can result in stator and rotor rubbing. Having a proper gap between stationary and rotating parts is core to ensuring the correct operating of the whole machine and to improve its life [1]. Labyrinth seals are currently the most common sealing methods used in the turbomachinery field since they have been introduced at the beginning of the previous century. The flow inside a labyrinth seal undergoes a series of consecutive constrictions and cavities as it is bounded by two relatively moving components, namely the teeth and the seal land. According to [2], two mechanisms occur inside a labyrinth; first, the flow accelerates as it approaches the clearance gap and a part of the pressure energy is converted into kinetic energy; then, part of the kinetic energy is dissipated in the cavity downstream of the constriction due to multiple phenomena, such as uncontrolled expansion, turbulence and viscous effects. The process repeats for each tooth of the labyrinth so that the flow resistance increases and hence, the control increases over the leakage flow rate. During the past decades, many parameters have been investigated in order to understand their influence on the labyrinth performance, such as pressure ratio [3, 4], Reynolds number [3], clearance gap [4, 5], tooth shape [6, 7] and rotational speed [8, 9]. It is clear that such a complexity should require some suitable CFD tools or a dedicated test campaign in order to evaluate both the flow field inside the seal cavities and the leakage flow rate. However, correlations can be employed to estimate the mass flow through the constrictions during the very first stages of the design process to obtain some useful information for the labyrinth dimensioning. Such a procedure saves a lot of time since it relies on some simple equations. In 1908, [PERSON] first studied the labyrinth glands as a series of discrete throttling passages acting similar to a series of orifices and proposed his well-known formula. Since then, many efforts have been made to develop new equations to carefully predict the leakage mass flow. Although each equation tries to model the same flow phenomena in different ways and with different levels of detail, all the existing models involve two parameters: a flow coefficient and a carry-over factor. The first one accounts for the effective amount of air which flows across the tip clearance due to the interaction with the boundary layer near the tooth tip; on the other hand, the carry-over factor considers the fact that the air inside the cavity is not completely brought to rest and hence, some kinetic energy is carried over to the following constriction. In general, direct and indirect approaches have been developed. The direct correlations employ lumped quantities which accounts for the behavior of the whole seal, without attempting to resolve the pressure drop between the different teeth. As a result, only one equation with a single calculation step is solved. On the contrary, the indirect approaches model the effects of each tooth by means of continuity equation across a single cavity, which allows us to determine the pressure distribution downstream each tooth. It is clear, therefore, that the reliability of the adopted correlative approach is crucial to the accuracy of the preliminary design phase, thus influencing, in turn, the speed and accuracy of the following detailed design. Therefore, the validation of different correlative approach, using detailed experimental data, is a relevant aspect of labyrinth seals research.