A mathematical model for the interaction of anisotropic turbulence with porous surfaces

Leading-edge noise is a complex phenomenon that occurs when a turbulent fluid encounters a solid object, and is a notable concern in various engineering applications. This study enhances a mathematical leading-edge noise model (Hales et al. , J. Fluid Mech. , vol. 970, 2023, A29) for anisotropic flow and porous boundaries. The model has two key components. First, we adjust the velocity spectrum to account for the possibility of anisotropy in the flow. This paper rigorously introduces a third dimension for the turbulence spectrum that preserves the turbulence kinetic energy and mathematical definitions for integral length scales. Second, we adapt the fully analytical acoustic transfer function to account for different boundaries by implementing convective impedance boundary conditions when formulating the gust-diffraction problem. This problem is then solved using the Wiener–Hopf technique. We discuss important aspects of this method, including the factorisation of a non-trivial scalar kernel function and the application of suitable edge conditions for the problem. Each modification is inspired by experimental leading-edge noise data using a series of different porous leading edges and anisotropic turbulence generated by a cylinder upstream of the edge. Experimental data demonstrate the interplay between anisotropy and leading-edge modifications while achieving the characteristic mid-frequency noise reduction expected from porous leading edges. Our model is adapted to best fit the trends of the data via a tailored impedance function, leading to good agreement with all datasets across an extended frequency range. This tailored function is used to successfully validate the model against other datasets from a different set of experiments.