Study of the attenuation of waves propagating through fixed and rotating turbine blades

Abstract

This paper deals with the prediction of the transmission and reflection coefficients of acoustic and entropy waves through a turbine. The two-dimensional analytical method of Cumpsty and Marble 5 for low frequencies, or compact row assumption, is extended to consider the enthalpy jump through a rotating blade. Numerical simulations of a rotor blade are used to validate the compact hypothesis and the extension of the theory. It is found that results are strongly dependent on the trailing edge condition used by Cumpsty and Marble 5 for the stator vane. Finally, the analytical method is used to reproduce the experimental data of Doyle and Matta. 7 Noise generated in the combustion chamber or by turbine blades propagates to the outlet of the engine through the downstream stator vanes and rotor blades. This propagation attenuates the acoustic emissions in the outlet, and should be thus considered when calculating combustion noise. At the same time, the acceleration and deceleration of entropy waves generated in the combustion chamber through non-uniform flows generates indirect noise, as shown by Marble and Candel, 16 which can contribute significantly to the total noise. 3,19,21 The propagation of acoustic and entropy waves through one-dimensional nozzle configurations has been studied in depth: Marble and Candel 16 proposed an analytical method to calculate the propagation of acoustic and entropy waves in a quasi-1D nozzle, based on the compact nozzle hypothesis. This hypothesis states that the wavelength of the perturbations is large compared to the axial extension of the nozzle, limiting the results to the low frequency range. Using the compact nozzle hypothesis the propagation of waves can be considered quasi-static and matching conditions can be written between both sides of the nozzle, corresponding to the mass, enthalpy and entropy conservation in a quasi-static regime. This analytical method can be used to calculate the attenuation of noise when acoustic waves propagate through a nozzle and at the same time to calculate the generation of indirect noise. Leyko et al. 15 used this analytical method to show that indirect combustion noise is relevant when considering typical aero-engine’s operational conditions. Experiments performed by Bake et al., 2 where an entropy wave was propagated through a nozzle, were explained both for the supersonic case with a shock wave 14 and for the subsonic case 8 using the compact nozzle hypothesis. The compact nozzle hypothesis has been extended both numerically 1,11 and analytically using asymptotic expansion methods. 9,24 Cumpsty and Marble 5 extended the one-dimensional analytical method to a fixed blade row. This extension takes into account the flow deviation imposed by the turbine blades and the cut-off frequencies of oblique acoustic modes present in real configurations. The method, based on the same compact hypothesis, uses similarly the mass, enthalpy and entropy equations to write the corresponding matching conditions. A fourth equation is obtained imposing the Kutta condition at the outlet of the blade. Using a matrix notation, the stages can be chained to obtain the transfer functions of a complete turbine. Recently, numerical simulations have been used to check the compact model and to analyse the errors made when considering