Three-dimensional modelling of serrated trailing-edge noise based on the Wiener-Hopf technique

Abstract

In this paper, a semi-analytical model based on the Wiener-Hopf technique is developed to predict the turbulent boundary layer trailing edge noise from serrated edges, aiming to account for the correct three-dimensional noise source and propagation effects. The scattered surface pressure over a semi-infinite flat plate is first obtained using the Green's function developed for the acoustic scattering by a serrated edge. A radiation integral over the flat plate of a finite size is subsequently performed to obtain the far-field noise using Amiet's approach, capturing the correct three-dimensional source and propagation effects. The model is subsequently validated by comparing it against the two-dimensional Wiener-Hopf-based model under various serration sizes and frequencies. Far-field spectral predictions show close agreement between the three- and two-dimensional models at moderate observer distances around $r/c=1$, where $r$ and $c$ represent the observer distance and the chord length, respectively. However, unlike the two-dimensional model, the present model successfully captures the far-field $1/r$ decay in noise amplitudes. In addition, the predicted directivity agrees well with the two-dimensional model at most observer angles, but also captures the correct dipolar behaviour at upstream angles and additional high-frequency lobes due to interference patterns induced by the finite flat plate. Compared to the previous three-dimensional serrated models, the present model is based on the Wiener-Hopf technique and achieves a speed-up ratio of two orders of magnitude. It is hoped that such a model may be used to enable an efficient numerical optimisation of the serration shape in realistic applications.