Abstract

Resonances are of importance in many fields of physics and engineering. In this lecture we limit ourselves to acoustic resonances in domains which are open to infinity in at least one direction and are excited mainly by unstable shear layers. Using perfectly matched layer absorbing boundary conditions in the form of the complex scaling method of atomic and molecular physics to approximate the radiation condition the resonance problem is transformed into a large eigenvalue problem which is solved numerically. Of particular interest are resonances with zero radiation loss (trapped modes) or very small radiation loss (nearly trapped modes). Such trapped modes can enhance or even dominate the shear layer instabilities causing high intensity tonal noise or structural damage. In domains which are open on all sides resonances are damped due to radiation losses. In laterally bounded domains, such as finite length structures or cavities in channels or pipes, trapped modes with zero radiation loss are possible. A well known example are the Parker modes in compressor cascades. Special attention will be paid to trapped modes in single and tandem plate cascades in an annular duct modelling the situation in axial flow compressors. All resonances are computed for zero mean flow approximating low Mach number flows. The dependence of the resonant frequency on various cascade parameters, such as blade length, blade number, blade stagger, blade sweep or gap between cascades is demonstrated.