Conceptual Flutter Analysis of Labyrinth Seals Using Analytical Models: Part II — Physical Interpretation

Abstract

A simple non-dimensional model to describe the flutter onset of labyrinth seals is presented. The linearized equations for a control volume which represents the inter-fin seal cavity, retaining the circumferential unsteady flow perturbations created by the seal vibration, are used. Firstly, the downstream fin is assumed to be choked, whereas in a second step the model is generalized for unchocked exit conditions. An analytical expression for the non-dimensional work-per-cycle is derived. It is concluded that the stability of a two-fin seal, depends on three non-dimensional parameters, which allow explaining seal flutter behaviour in a comprehensive fashion. These parameters account for the effect of the pressure ratio, the cavity geometry, the fin clearance, the nodal diameter, 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 non-dimensional 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 [1] and Abbot [2], 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 inter-fin cavity gives rise to a correction of the resonance frequency and shifts the stability region. The non-dimensionalization 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[3] detailed the derivation of the theoretical model and drew some preliminary conclusions. Part II analyzes in depth the implications of the model and outlines the extension to multiple cavity seals.