Resonant acoustic frequencies of a tandem cascade. Part 1. Zero relative motion

In this paper we study the acoustic scattering between two flat-plate casca des, with the aim of investigating the possible existence of trapped modes. In practical terms this question is related to the phenomenon of acoustic resonance in turbomachinery, whereby such resonant modes are excited to lar ge amplitude by unsteady processes such as vortex shedding. We use the Wien er-Hopf technique to analyse the scattering of the various wave fields by t he cascade blades, and by considering the fields between adjacent blades, a s well as between the cascades, we are able to take full account of the gen uinely finite blade chords. Analytic expressions for the various scattering matrices are derived, and an infinite-dimensional matrix equation is forme d, which is then investigated numerically for singularity. One advantage of this formulation is that it allows the constituent parts of the system to be analysed individually, so that for instance the behaviour of the gap bet ween the blade rows alone can be investigated by omitting the finite-chord terms in the equations. We demonstrate that the system exhibits two types o f resonance, at a wide range of parameter values. First, there is a cut-on/ cut-off resonance associated with the gap between the rows, and correspondi ng to modes propagating parallel to the front face of the cascades. Second, there is a resonance of the downstream row, akin to a Parker mode, driven at low frequencies by a vorticity wave produced by trapped duct modes in th e upstream row, and at higher frequencies by radiation modes land the vorti city wave) between the blade rows. The predictions for this second set of r esonances are shown to be in excellent agreement with previous experimental data. The resonant frequencies are also seen to be real fdr this twin casc ade system, indicating that the resonances correspond to genuine trapped mo des. The analysis in this paper is completed with non-zero axial flow bur w ith zero relative rotation between the cascades - in Part 2 (Woodley & Peak e 1999) we will show how non-zero rotation of the upstream row can be inclu ded.