Acoustic and shock wave dispersion in cylindrical ducts

Abstract

An asymptotic approach is employed to investigate the effects of dispersion on acoustic disturbances propagating through cylindrical ducts, the walls of which may be impermeable or acoustically treated. The disturbances are assumed to be those generated by a rotor-locked pressure field. In the large azimuthal wavenumber (high-frequency) limit, m ≫ 1, linear potential perturbations are found to be weakly dispersive, with the leading order correction occurring at O(m−2/3). When a liner is present, its additional dispersive and dissipative effects appear at O(m−1). The effects of weak nonlinearity, in addition to those of dispersion are then studied. Specifically, on the assumption that the disturbance is comprised of a single radial mode, an evolution equation is derived for the pressure field. Numerical solutions of this equation are carried out for idealized perturbation patterns corresponding to tuned and detuned rotor blades, featuring respectively shocks of uniform and varying strength. In the tuned, untreated case, dispersive effects are found to reduce the dissipation across the shocks without substantially changing the character of the waveform in comparison with the non-dispersive case. The inclusion of the acoustic liner brings about more drastic changes: the periodic shock pattern is transformed into a series of humps with dispersive tails. With the detuned blades, it is found that the untreated duct scenario yields a waveform that differs significantly from the non-dispersive case. In particular, the shock merging that occurs in the latter instance does not transpire. Instead, the shocks exchange amplitudes even as they decay. The presence of the acoustic liner in the detuned case is similar to the tuned case, except with a richer spectrum being generated. Despite the relative simplicity of the models studied, these results exhibit many of the features of previous numerical simulations and point to the need for the inclusion of dispersive effects in designing liners for turbofan applications.