A simple nondimensional model to describe the flutter onset of labyrinth seals is presented. The linearized mass and momentum integral equations for a control volume which represents the interfin seal cavity, retaining the circumferential unsteady flow perturbations created by the seal vibration, are used. First, the downstream fin is assumed to be choked, whereas in a second step the model is generalized for unchoked exit conditions. An analytical expression for the nondimensional work-per-cycle is derived. It is concluded that the stability of a two-fin seal depends on three nondimensional parameters, which allow explaining seal flutter behavior in a comprehensive fashion. These parameters account for the effect of the pressure ratio, the cavity geometry, the fin clearance, the nodal diameter (ND), the fluid swirl velocity, the vibration frequency, and the torsion center location in a compact and interrelated form. A number of conclusions have been drawn by means of a thorough examination of the work-per-cycle expression, also known as the stability parameter by other authors. It was found that the physics of the problem strongly depends on the nondimensional acoustic frequency. When the discharge time of the seal cavity is much greater than the acoustic propagation time, the damping of the system is very small and the amplitude of the response at the resonance conditions is very high. The model not only provides a unified framework for the stability criteria derived by Ehrich (1968, “Aeroelastic Instability in Labyrinth Seals,” ASME J. Eng. Gas Turbines Power, 90(4), pp. 369–374) and Abbot (1981, “Advances in Labyrinth Seal Aeroelastic Instability Prediction and Prevention,” ASME J. Eng. Gas Turbines Power, 103(2), pp. 308–312), but delivers an explicit expression for the work-per-cycle of a two-fin rotating seal. All the existing and well-established engineering trends are contained in the model, despite its simplicity. Finally, the effect of swirl in the fluid is included. It is found that the swirl of the fluid in the interfin cavity gives rise to a correction of the resonance frequency and shifts the stability region. The nondimensionalization of the governing equations is an essential part of the method and it groups physical effects in a very compact form. Part I of the paper details the derivation of the theoretical model and draws some preliminary conclusions. Part II of the corresponding paper analyzes in depth the implications of the model and outlines the extension to multiple cavity seals.

A simple nondimensional model to describe the flutter onset of two-fin straight labyrinth seals (Corral and Vega, 2018, “Conceptual Flutter Analysis of Labyrinth Seals Using Analytical Models—Part I: Theoretical Background,” ASME J. Turbomach., 140(10), p. 121006) is extended to stepped seals. The effect of the axial displacement of the seal is analyzed first in isolation. It is shown that this fundamental mode is always stable. In a second step, the combination of axial and torsion displacements is used to determine the damping of modes with arbitrary torsion centers. It is concluded that the classical Abbot's criterion stating that seals supported on the low-pressure side of the seal are stable provided that natural frequency of the mode is greater than the acoustic frequency breaks down under certain conditions. An analytical expression for the nondimensional work-per-cycle is derived and new nondimensional parameters controlling the seal stability identified. It is finally concluded that the stability of stepped seals can be assimilated to that of a straight through seal if the appropriate distance of the torsion center to the seal is chosen.

A simple non-dimensional model to describe the flutter onset of two-fin straight labyrinth seals [1] is extended to account for non-isentropic flow perturbations. The isentropic relationship is replaced by the more general integral energy equation of the inter-fin cavity. A new expression for the Corral & Vega stability criterion is derived which is very consistent with the previous model in the whole design space of the seal but for torsion centers located in the high-pressure side close to the seal. The new model formally depends on more dimensionless parameters since the existing parameter grouping of the previous model does not hold anymore, but this dependency is weak in relative terms. The model blends the limit where the discharge time of the inter-fin cavity is much longer than the vibration period, and the flow is nearly isentropic, and the opposite limit, where the perturbations are isothermic, gracefully. A few numerical examples obtained using a three-dimensional linearized frequency domain solver are included to support the model and show that the trends are correct. The matching between the work-per-cycle obtained with the model and frequency domain solver is good. It is shown that some weird trends obtained using linearized unsteady simulations are qualitatively consistent with the current model but not with the previous one [1]. The largest differences between the new and the previous model are seen when the seal is supported at the high-pressure side.

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