This paper is predominantly an experimental study into the reduction of turbulence -aerofoil interaction noise by the introduction of aerofoil porosity. In this paper we study three scenarios applied to flat plates: (a) when the flat plate is fully porous, (b) when the flat plate is partially porous from the leading edge and (c) when porosity is introduced downstream of the leading edge.This paper shows that the noise reduction spectra collapse when plotted against non-dimensional frequency f l/U , where l is the length of porous section and U is the flow velocity. Narrow band measurements on a partially porous aerofoil have shown that its noise reduction spectra is characterised by a number of narrow peaks. This paper proposes two main mechanisms for explaining this behaviour. The noise reduction mechanisms are validated against noise reductions measured on a realistic aerofoil at relatively low angles of attack. One of the key findings of this paper is that, by using only a single row of holes downstream of the aerofoil leading edge one can obtain significant levels of noise reduction.

This paper presents a generalization of a recent iterative approach to solving a class of 2 × 2 matrix Wiener–Hopf equations involving exponential factors. We extend the method to square matrices of arbitrary dimension n , as arise in mixed boundary value problems with n junctions. To demonstrate the method, we consider the classical problem of scattering a plane wave by a set of collinear plates. The results are compared to other known methods. We describe an effective implementation using a spectral method to compute the required Cauchy transforms. The approach is ideally suited to obtaining far-field directivity patterns of utility to applications. Convergence in iteration is fastest for large wavenumbers, but remains practical at modest wavenumbers to achieve a high degree of accuracy. This article is part of the theme issue ‘Modelling of dynamic phenomena and localization in structured media (part 2)’.

This paper considers the use of compliant boundary conditions to provide a homogenized model of a finite array of collinear plates, modelling a perforated screen or grating. While the perforated screen formally has a mix of Dirichlet and Neumann boundary conditions, the homogenized model has Robin boundary conditions. Perforated screens form a canonical model in scattering theory, with applications ranging from electromagnetism to aeroacoustics. Interest in perforated media incorporated within larger structures motivates interrogating the appropriateness of homogenized boundary conditions in this case, especially as the homogenized model changes the junction behaviour considered at the extreme edges of the screen. To facilitate effective investigation we consider three numerical methods solving the Helmholtz equation: the unified transform and an iterative Wiener–Hopf approach for the exact problem of a set of collinear rigid plates (the difficult geometry of the problem means that such methods, which converge exponentially, are crucial) and a novel Mathieu function collocation approach to consider a variable compliance applied along the length of a single plate. We detail the relative performance and practical considerations for each method. By comparing solutions obtained using homogenized boundary conditions to the problem of collinear plates, we verify that the constant compliance given in previous theoretical research is appropriate to gain a good estimate of the solution even for a modest number of plates, provided we are sufficiently far into the asymptotic regime. We further investigate tapering the compliance near the extreme endpoints of the screen and find that tapering with $\tanh $ functions reduces the error in the approximation of the far field (if we are sufficiently far into the asymptotic regime). We also find that the number of plates and wavenumber has significant effects, even far into the asymptotic regime. These last two points indicate the importance of modelling end effects to achieve highly accurate results.

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This paper presents a semi-analytical investigation into the aeroacoustic properties of aerofoils with porous leading edges, complemented by an experimental study of flat rigid plates with a perforated leading section. Interest in partially porous aerofoils arises from their potential to reduce noise emission from wind turbines and commercial aeroplanes whilst maintaining aerodynamic performance. Developing theoretical models will be helpful for understanding the dominant physical mechanisms and thereby designing optimal implementations. We consider a simple theoretical model that captures acoustic scattering from a finite plate with a porous leading edge. This yields a matrix Wiener-Hopf problem which is solved by an iterative approach that facilitates reliable and rapidly computable solutions. This allows a quantification of acoustic reductions that may be anticipated, and a comparison between studying a porous leading edge on a finite and semi-infinite plate. Experimental results are presented for a flat rigid plate with a perforated leading edge section that demonstrate some qualitative agreement with the theoretical model.