Do we need non-linear corrections? On the boundary Forchheimer equation in acoustic scattering

This paper presents a rapid numerical method for predicting the aerodynamic noise generated by foam-like porous aerofoils. In such situations, particularly for high-frequency noise sources, Darcy’s law may be unsuitable for describing the pressure jump across the aerofoil. Therefore, an inertial Forchheimer correction is introduced. This results in a non-linear boundary condition relating the pressure jump across the material to the fluid displacement. We aim to provide a quick, semi-analytical model that incorporates such non-linear effects without requiring a full turbulent simulation. The numerical scheme implemented is based on local Mathieu function expansions, leading to a semi-analytical boundary spectral method that is well-suited to both linear and non-linear boundary conditions (including boundary conditions more general than the Forchheimer correction). In the latter case, Newton’s method is employed to solve the resulting non-linear system of equations for the unknown coefficients. Whilst the physical model is simplified to consider just the scattering by a thin porous aerofoil with no background flow, when the non-linear inertial correction is included good agreement is seen between the model predictions and both experimental results and large eddy simulations. It is found that for sufficiently low-permeability materials, the effects of inertia can outweigh the noise attenuation effects of viscosity. This helps explain the discrepancy between experimental results and previous (linear) low-fidelity numerical simulations or analytical predictions, which typically overestimate the noise reduction capabilities of porous aerofoils.