Asymptotic approach to aeroacoustics
The present thesis is concerned with mechanisms of sound generation by unsteady hydrodynamic motions in shear flows. Such problems are of great importance for the aviation industry, where noise reduction has long been a serious concern. The current state of the art in this field primarily rests on the acoustic analogy theory initiated by Lighthill (1952) and its numerous variants. An alternative method, based on matched asymptotic expansions, was developed by Crow (1970) among others. It was recently further advanced further within the framework of triple-deck theory to predict sound emission from certain viscous flows in the presence of a boundary (Wu 2002). We apply this approach to three aeroacoustic problems, which are deemed suitable for a triple-deck approach, with a view to analyze some fundamental sound generation processes on a self-consistent first-principle basis. The first problem addresses sound generation in a transonic boundary layer subjected to unsteady suction and injection. The analogous problem was analysed by Wu (2002) for subsonic flows but the theory breaks down at transonic speeds. The transonic effect first manifests itself when 1 −M2 = O(R−1/9), where M is the free-stream Mach number and R the global Reynolds number, which is assumed large. The unsteady flow due to the suction/injection is accommodated by a somewhat different triple-deck structure, in which the unsteady effect appears at the leading order in the upper deck, as was found by Bowles & Smith (1993). It is found that the resulting sound field is fundamentally different from its subsonic counterpart. The subsonic flow field involves an outer region, in which the pressure assumes an acoustic character. Such an outer region is not present in the transonic case, and it is not possible to express the solution in terms of multipoles since the source is not compact. Most importantly, it is found that the radiated sound produces a leading-order ‘back action’ effect on the source. The second problem is concerned with the acoustic radiation emitted by instability waves as they undergo rapid distortion, which has been recognized as one of prime mechanisms by which instability waves generate sound. We consider the situation where the rapid distortion in the Tollmien-Schlichting (T-S) waves is caused by a localized surface roughness in a compressible subsonic boundary layer. We find that in order to predict the leading-order acoustic pressure fluctuation, the first four terms in the expansion for the hydrodynamic field have to be determined. They contribute equally to the radiated sound because they act as octupole, quadrupole, dipole and monopole sources respectively. The analysis reveals two types of cancellations, which may explain the difficulties in accurately predicting aerodynamic sound. The first occurs in spectral space in the small-wavenumber limit; this cancellation renders the leading-order source to act as an octupole rather than a quadrupole source. The second type occurs in physical space, among the sources in different regions of the flow. The analysis also shows that a localized roughness influences the energetics of a T-S wave and this effect can be characterized by a transmission coefficient. The third problem that we analyze is that of sound emission due to a stationary source embedded in a boundary layer. Our aim is to obtain some generic results, which generalise the studies of individual cases to encompass a broad class of flows. For that purpose, we seek Green’s functions corresponding to an arbitrarily specified source located in the main deck. The temporal and spatial scales of the source are assumed to be compatible with those of triple-deck theory so that the near-field hydrodynamics can be described by a triple-deck structure and a fourth, outer region accommodates the acoustic pressure. The solution is sought in terms of an asymptotic series for the pressure. The required order of approximation depends on the radiating nature of the effective source q, which can be characterized by its behaviour in the small wavenumber k limit. If q = O(1) for k = 0, the first two terms are needed. They act as dipole and monopole sources respectively and so contribute equally to the sound in an asymptotic sense. If q = O(k) for k 1, the first three terms have to be obtained, which act as quadrupole, dipole and monopole sources respectively.