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

TL;DR

This work develops a three-dimensional semi-analytical model for serrated trailing-edge noise using a Wiener-Hopf Green's function to obtain near-field surface pressure on a serrated edge and a Curle/Amiet-type radiation integral for the far field. It extends prior two-dimensional Wiener-Hopf models by correctly predicting the $1/r$ far-field decay and three-dimensional directivity associated with finite plate span and chord, while delivering a two-order-of-magnitude speed-up over Schwarzschild-based methods. Validation shows good agreement with a 2D model at moderate observer distances but reveals distinctive 3D features such as upstream dipolar attenuation and interference-induced lobes, particularly for sharper serrations. The framework enables efficient serration-geometry optimization and can be extended to general piecewise-linear serrations under a frozen turbulence assumption, offering practical impact for rotorcraft and wind-turbine noise design.

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.

Figures (7)

• Figure 1: The simplified model of a flat plate with sawtooth serrations positioned in a uniform streamwise flow. The chord and span lengths of the plate are $c$ and $d$, respectively, while the wavelength and root-to-tip amplitude of the serration are denoted by $\lambda$ and $2h$, respectively. The coordinates $y_1, y_2, y_3$ are aligned with the streamwise, spanwise, and normal-to-plate directions, respectively. The far-field observer is located at $(x_1, x_2, x_3)$.
• Figure 2: The polar coordinates used in the derivation. $(r,\theta)$ is the polar coordinate frame in the stretch $y_1/\beta-y_3$ plane. Two auxiliary polar coordinate frames originate from $(h/\beta,0)$ and $(-h/\beta,0)$, respectively.
• Figure 3: Distribution of the real part of the scattered pressure $R_s$ in equation \ref{['eqn:R_s']}. $R_s$ is resulted from a wall pressure gust with $K_2=0$, $M_0=0.1$ and $h=1/3$.
• Figure 4: Comparison of the PSD of the predicted noise from the two- and three-dimensional models for serrated and straight trailing edges. The observer at $\theta = 90^\circ$ and $x_3/c=1$ above the plate in the mid-span plane with $M_0=0.1$: (a) $\lambda/h=8, h/c=0.025$; (b) $\lambda/h=4, h/c=0.025$; (c) $\lambda/h=2, h/c=0.05$; (d) $\lambda/h=1, h/c=0.005$; (e) $\lambda/h=0.4, h/c=0.05$; (f) $\lambda/h=0.2, h/c=0.05$.
• Figure 5: Comparison of the PSD of the predicted noise from the two- and three-dimensional models for serrated and straight trailing edges. The observer at $\theta = 90^\circ$ and $x_3 = r$ above the plate in the mid-span plane with $M_0=0.1$, $\lambda/h=0.4$, and $h/c=0.1$. (a) $kc=1$; (b) $kc=10$.