Aerodynamic noise from rigid trailing edges with finite porous extensions

Abstract

This paper investigates the effects of finite flat porous extensions to semi-infinite impermeable flat plates in an attempt to control trailing-edge noise through bio-inspired adaptations. Specifically the problem of sound generated by a gust convecting in uniform mean steady flow scattering off the trailing edge and the permeable–impermeable junction is considered. This set-up supposes that any realistic trailing-edge adaptation to a blade would be sufficiently small so that the turbulent boundary layer encapsulates both the porous edge and the permeable–impermeable junction, and therefore the interaction of acoustics generated at these two discontinuous boundaries is important. The acoustic problem is tackled analytically through use of the Wiener–Hopf method. A two-dimensional matrix Wiener–Hopf problem arises due to the two interaction points (the trailing edge and the permeable–impermeable junction). This paper discusses a new iterative method for solving this matrix Wiener–Hopf equation which extends to further two-dimensional problems, in particular those involving analytic terms that exponentially grow in the upper or lower half-planes. This method is an extension of the commonly used ‘pole removal’ technique and avoids the need for full matrix factorisation. Convergence of this iterative method to an exact solution is shown to be particularly fast when terms neglected in the second step are formally smaller than all other terms retained. The new method is validated by comparing the iterative solutions for acoustic scattering by a finite impermeable plate against a known solution (obtained in terms of Mathieu functions). The final acoustic solution highlights the effects of the permeable–impermeable junction on the generated noise, in particular how this junction affects the far-field noise generated by high-frequency gusts by creating an interference to typical trailing-edge scattering. This effect results in partially porous plates predicting a lower noise reduction than fully porous plates when compared to fully impermeable plates.