Acoustic impedance of a quarter-wave resonator with non-uniform temperature

This study proposes a semi-analytical approach to predict the acoustic impedance of a quarter-wave resonator with non-uniform temperature. In analogy to well-known models for Helmholtz resonators, the quarter-wave resonator is represented as a mass-spring-damper system. The treatment is limited to polynomial temperature profiles, but does take into account that the resonator cavity is not acoustically compact. In order to assess the validity and accuracy of the semi-analytical results, computational fluid dynamics (CFD) simulations are carried out for a variety of temperature profiles in a quarter-wave resonator cavity. The acoustic reflection offered by the resonator opening to imposed incoming acoustic waves is evaluated, either by spectral analysis of time series generated by repeated, monofrequent excitation, or by system identification of acoustic signals generated with broad-band excitation. System identification - a variant of supervised machine learning - requires only a single simulation run to characterize the resonator over the frequency range of interest. The validation study shows good quantitative agreement between the mono-frequent and broadband excitation cases, as well as qualitative consistency with the analytical predictions. The present study demonstrates that resonator eigenfrequencies as well as maximum effectiveness and bandwidth of acoustic damping are quite sensitive to temperature inhomogeneities. In the context of thermo-acoustic combustion instability, where resonators are frequently employed as a means of passive control. The results suggest that the acoustic characteristics of a resonator with hot combustion products at the inlet and a cooled backing cannot be computed simply with a representative average of the temperature distribution. These findings underline the necessity of a comprehensive design process, which includes thermal analysis in order to assure optimum resonator effectiveness.