Conceptual Flutter Analysis of Labyrinth Seals Using Analytical Models—Part I: Theoretical Support

Abstract

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.

Numerical Validation of an Analytical Seal Flutter Model

“…which is a generalisation of the expression derived in Corral and Vega (2018). The limiting value ofẽh below which seals supported on the HPS are always stable oscillates betweenẽ e f f h = −1 for very low pressure ratios (h → ∞) andẽ e f f h −0.89 for choked seals (h = 2.5).…”

“…The higher order formulation of the CV model is described in is an extension of the original model (Corral and Vega, 2018) to account for non-isentropic flow perturbations. In this formulation, the entropy conservation relationship is replaced by the integral energy equation of the seal inter-fin cavity.…”

“…The dimensionless work-per-cycle depends on four nondimensional parameters where the non dimensional parameterẽ = γrH/(sL) includes the effect of the vibration mode-shape, the seal clearance and the geometry of the seal. The expression of the out of phase component of the unsteady pressure is formally equal to that derived in the isentropic formulation (Corral and Vega, 2018) where the seal clearance non-dimensional parameter appears always multiplied by h which is a function of the seal pressure ratio. Nevertheless in the higher order formulation the parameterẽh is replaced byẽh e f f =ẽh F(Ω) to account for non-isentropic perturbations.…”

Higher-Order Conceptual Model for Labyrinth Seal Flutter

“…A few studies on this subject were conducted in the 60's and 70's, but until recently, the understanding of labyrinth seals has been mostly based on the conclusions drawn in these pioneering works. Recently, a simple physics based conceptual model has been derived to increase the understanding of straight seal instabilities [1,10] and new light to the problem has been shed. This model showed the existence of new dimensionless parameters which played a key role in seal instability.…”

“…Corral and Vega [1,10] have recently proposed a new comprehensive conceptual model for seal flutter derived from first principles. The model not only conciliates the observations of Ehrich [16] and Abbot [17], but provides a more refined but still simple stability limit.…”