Acoustic liner - mean flow interaction

Since the introduction in the 1950’s of the jet engine in civil aviation, its noise has been a stringent problem. National laws and international regulations forbid selling noisy aircraft and limit the total yearly noise load of airports, this noise load depending on both the number of flight motions (starts and landings), and the noise emitted per aircraft. Since the number of flight motions increases in direct proportion to economic growth, the noise per aircraft has to decrease in compensation. As a result, year by year a world wide effort in aeroacoustic research remains necessary. From elementary dimensional arguments, scaling is hardly possible, while full scale experiments are very expensive, thus making almost any mathematical model cheaper and better. We study in this thesis aspects of parts of the mathematical model used in duct acoustics, and focus on the interaction of acoustic liners with the mean flow, discussing reliability and accuracy in this context. A more than 50 years old modeling problem was the correctness of a vanishing boundary layer along an acoustic liner. Arguing classically, the acoustic effects of a boundary layer that is much thinner than any characteristic wave length is the same as of a vanishing boundary layer. So for a numerically e??cient and thrifty model without unnecessary parameters, it is reasonable to apply this limit yielding the so-called Ingard-Myers condition. This works well in frequency domain. Our research showed that in time domain, on the other hand, a boundary layer less than a certain (very small, but non-zero) thickness is absolutely unstable. This makes the model useless for any industrially relevant configuration. We propose here a corrected version of the limit that retains the stability properties of the finite boundary layer. In addition to this, we give an estimate for the critical boundary layer thickness, beyond which the flow is absolutely unstable. The last part of the thesis discusses the critical layer singularity arising in the mathematical model due to the inviscid assumption. Common treatment is to by-pass its contribution assuming it is negligible, without fully understanding its subtleties. We study this problem for linear-shear boundary layers over acoustic linings. We show that if the source is located in the boundary layer, neglecting the critical layer means neglecting also the trailing vorticity produced by source, which introduces significant errors. Moreover, extra care is needed for high frequencies, due to an existing leaking mode.