Mathematical Modelling of Acoustic Diffraction Noise Embracing Diverse Boundary Conditions

Abstract

In this thesis, we develop advanced mathematical tools for modelling acoustic diffraction noise in various contexts and explore improved models for aerofoil-turbulence interaction noise. The thesis is divided into two parts. The first explores variants of half-plane scattering problems and the analytical requirements for their solutions. The second focuses on applying solutions of these scattering problems to leading- and trailing-edge noise models and developing a mathematical framework for the statistical descriptions of turbulence within these models. Part I, "Applying the Wiener–Hopf Technique to Diverse Diffraction Problems in Acoustics", develops a generalised theory to solve diffraction problems with linear boundary conditions prescribed distinctly on each side of a semi-infinite boundary. We choose this framework to reflect physical applications in subsequent parts. Key aspects covered in the technique include the multiplicative splitting of polynomial kernels within Wiener–Hopf equations, adapting edge conditions to more mathematically involved boundary conditions, and solving two-sided diffraction problems with distinct boundary conditions on each side, leading to reworkings of the Wiener–Hopf technique with intriguing results. Part II, "Adapting Leading- and Trailing-Edge Noise Models to Anisotropy and Compliant Plates", focuses on creating a generalised framework to predict aerofoil turbulence interaction noise. This noise arises when turbulent flow scatters off an aerofoil’s sharp leading or trailing edge. Existing analytical models use the Wiener–Hopf technique, approximating turbulence as a single turbulent eddy and scattering it off an edge modelled as a zero-thickness semi-infinite boundary with an infinite span. These solutions enable the construction of a transfer function that relates the incident gust to its far-field scattered pressure. We sum the solutions for all possible eddies using a turbulence spectrum that associates each eddy with its expected energy and length scales. This approach is valid for both leading and trailing-edge noise models. Slight changes in turbulence modelling are addressed carefully throughout this part of the thesis. The scattering component of both models requires theory developed in the first part of the thesis. We verify our leading-edge model with experimental work exploring the interaction of anisotropic turbulence with a rigid leading edge. Then, we repeat the experiment twice more to investigate different types of edges. First, we place a layer of noisereducing foam on either side of the leading edge. Second, we use spanwise-perforated inserts. All experiments show promise for approaching analytical modelling of leading edges using an analytical transfer function based on the solution of the interaction of a gust with a semi-infinite plate with an impedance boundary condition that accounts for steady mean flow effects. Finally, our focus on trailing-edge noise is twofold. First, we predict how changing the trailing edge’s material properties (impedance) can affect the far-field noise. Second, we conduct a theoretical investigation into an analytical pressure spectrum: the TNO–Blake model. We discuss how some analytical simplification can be altered or improved to apply to a broader range of contexts in future studies into either pressure spectra or their use within analytical trailing-edge noise models.