Continuous Adjoint to Proudman's Formula for Aeroacoustic Shape Optimization

This paper presents an approach for aeroacoustic optimization through the reduction of acoustic sources, based on the integration of Proudman's formula into a continuous adjoint framework coupled with the Reynolds-averaged Navier–Stokes equations, for the first-time. The development includes the adjoint to the $k-\omega SST$ turbulence model. Here, Proudman's formula is used to compute acoustic emissions of turbulent flows around aerodynamic bodies using the turbulent kinetic energy and specific rate of dissipation. Broadband noise generation through Proudman's formula is initially validated for a case including the flow around an isolated airfoil. Subsequently, the sensitivity derivatives of an objective function quantifying acoustic sources are verified against finite differences, with optimizations of two isolated airfoils and the MEXICO wind turbine following. Optimizations are conducted by extending the $adjointOptimisationFoam$ tool in OpenFOAM, developed and made publicly available by the group. During the optimization, constraints on the lift force, the drag force, the pitching moment coefficient, the torque, the trailing edge thickness, and airfoil volume are imposed, depending on the case. The geometries and grids are parameterized using PARSEC and morphing boxes based on volumetric B-Splines. The optimizations result in shapes with reduced acoustic sources while preserving aerodynamic efficiency, highlighting the effectiveness of the proposed method and programmed software.