Acoustic receptivity in compressible boundary layer flows over aerofoils

Abstract

The generation of streamwise instabilities through scattering of an acoustic wave by surface roughness, suction or heating is studied with a time-harmonic compressible adjoint linearised Navier-Stokes approach for subsonic flow conditions. The proposed acoustic receptivity model builds on the ideas of the Goldstein-Ruban theory and extends them to more complex geometries such as aerofoils. The adjoint methodology of Hill in the context of acoustic receptivity is applied to the compressible flow regime and an alternative formulation to predict sensitivity to the angle of incidence of an acoustic wave is proposed. A novel approach to model the acoustic field around an aerofoil is undertaken. This is based on a three-layer decomposition, namely the far-field solution, the inviscid wave-aerofoil interaction and the acoustic boundary layer. High Strouhal number analytical solutions to the compressible Stokes layer problem in flat plate geometries are deduced and shown to be in better agreement with numerical solutions compared to previous works.

The acoustic receptivity model detailed in this thesis is shown to be an accurate and efficient means of predicting receptivity amplitudes, and therefore to be more suitable for parametric investigations than other high-fidelity approaches. Comparison with direct numerical simulations and with the simpler finite Reynolds number approach provides strong evidence of the correctness of the approach, including the ability to quantify non-parallel flow effects. These effects are found to be negligible in zero pressure gradient flat plate geometries but to be sizeable under strong adverse pressure gradients and at high frequencies in Falkner-Skan boundary layers.

Often in the literature, for subsonic Mach numbers, acoustic receptivity studies are restricted to downstream travelling waves. In this work we remove the assumption that the acoustic wave travels in a single direction and investigate the validity of the linearised unsteady boundary layer equations (LUBLE) as a model of the acoustic boundary layer across the Mach number and angle of incidence parameter space for a flat plate. For near upstream travelling waves at high subsonic Mach numbers, the comparatively short wavelengths of the acoustic waves cause sizeable pressure oscillations across the boundary layer, thus invalidating the LUBLE. In such cases, the LUBLE should be replaced by a more accurate model of the acoustic signature within the boundary layer. We explore the use of the forced compressible linear stability equations, in full and in the high Reynolds number limit, as alternative models. While these models predict deviations from the LUBLE results that are in agreement with previous research in the literature, we find that they degenerate for waves travelling parallel to the flat plate.

Lastly, acoustic-roughness receptivity on a NACA0012 aerofoil is investigated. A parametric study of the influence of the acoustic wave angle of incidence and of the position and width of a localised roughness element on receptivity is undertaken. Non-trivial dependencies on the angle of incidence are observed; this is owed to complex wave patterns in the vicinity of the aerofoil due to a combination of reflection and diffraction of the wave by the obstacle.