Predicted Rotordynamic Behavior of a Labyrinth Seal as Rotor Surface Speed Approaches Mach 1

Prior one-control-volume (1CV) models for rotor-fluid interaction in labyrinth seals produce synchronously reduced (at running speed), frequency-independent stiffness and damping coefficients. The 1CV model, consisting of a leakage equation, a continuity equation, and a circumferential-momentum equation (for each cavity), was stated to be invalid for rotor surface speeds approaching the speed of sound. However, the present results show that while the 1CV fluid-mechanic model continues to be valid, the calculated rotordynamic coefficients become strongly dependent on the rotor’s precession frequency. A solution is developed for the reaction-force components for a range of precession frequencies, producing frequency-dependent stiffness and damping coefficients. They can be used to define a Laplace-domain transfer-function model for the reaction-force/rotor-motion components. Calculated results are presented for a simple Jeffcott rotor model acted on by a labyrinth seal. The model’s undamped natural frequency is 7.6 krpm. The fluid properties, seal radius $Rs$⁠, and running speed $ω$ cause the rotor surface velocity $Rsω$ to equal the speed of sound $c0$ at $ω=58krpm$⁠. Calculated synchronous-response results due to imbalance coincide for the synchronously reduced and the frequency-dependent models. For an inlet preswirl ratio of 0.5, both models predict the same log-dec out to $ω≈14.5krpm$⁠. The synchronously reduced model predicts an onset speed of instability (OSI) at 10 krpm, but a return to stability at 48 krpm, with subsequent increases in log-dec out to 70 krpm. The frequency-dependent model predicts an OSI of 10 krpm and no return to stability out to 70 krpm. The frequency-dependent models predict small changes in the rotor’s damped natural frequencies. The synchronously reduced model predicts large changes. The stability-analysis results show that a frequency-dependent labyrinth seal model should be used if the rotor surface speed approaches a significant fraction of the speed of sound. For the present example, observable discrepancies arose when $Rsω=0.26c0$⁠.