Analytical Prediction of Acoustic Resonance in Labyrinth Seals

Labyrinth seals are an essential and widely used component of modern turbomachines and can contribute significantly to their efficiency. Their often complex geometry is intended to generate pressure losses to reduce leakage flow. Acoustic resonances can occur within the cavities used for this purpose, which, when combined with a matched source of excitation, can result in high noise emissions and vibration excitation of the mechanical components of the labyrinth seal. In this work, an analytical model is developed and validated with numerical data to predict the resonant frequencies of axial standing waves within the cavities for propagating acoustic modes. The model is based on a RANS simulation of the cavity flow and determines the axial characteristics of the acoustic modes propagating upstream and downstream at several axial positions. Two acoustic wave equations commonly used in turbomachinery applications are used to analytically estimate the sound propagation within the seal's cavity. An overall axial wavelength across the cavity is estimated based on the wavenumbers determined at each axial position and a resonance condition is formulated to iterate the modal resonance frequency. This modeling approach is used to estimate the resonant frequency of an axial standing wave for different acoustic mode orders. Comparison of the models with numerical results in a generic labyrinth seal for three aerodynamic operating points shows a deviation between the analytically and numerically determined resonant frequency of approximately 0.68% on average with resonant frequencies occurring at Helmholtz numbers between 58.69 and 64.20 at Mach numbers of up to 0.4. Numerical setup studies regarding the influence of turbulence models, turbulent production limiters, and modeling of rotational effects on the turbulence in RANS simulations yielded in a deviation smaller than 1% of the numerically predicted resonance frequencies.