Equal-loudness-adaptive time integration for the high-order solution of direct aeroacoustics simulations

In the last years, numerical simulations have become a routine tool for the acoustic predictions in many industrial fields, e.g., sound engineering, architecture acoustics, and noise control. Aeroacoustic simulations are characterized by a wide range of temporal scales, and need a high accuracy in both time and space with a robust, accurate, and efficient long-time integration. The discontinuous Galerkin finite element spatial discretization is well suited for this class of simulations, both for computational fluid dynamics and aeroacoustics, thanks to the high-order accuracy. An efficient time integration is of fundamental importance to overcome the weakness of these solvers, which is the high computational cost, and can be achieved by an appropriate combination of high-order time integration schemes and algorithms to adapt the time step-size. Information about the robustness and accuracy of time step-size adaptation algorithms is only available in literature for fluid dynamics simulations. The aim of this work is to propose and validate an algorithm to adapt the time step-size for aeroacoustic simulations, where the user-defined adaptive tolerance is replaced with the physical value of the sound pressure level from an acceptable equal-loudness-level calculated with the reference ISO 226:2023(E). The efficiency, in terms of accuracy and robustness, of the proposed time integration is proved with the simulations of the laminar flow around a circle, airfoil, and quadrangle, with different solution approximations, control points, flow conditions, initial adaptive tolerances, and equal-loudness-levels. The time step-size is increased or decreased according to the initial adaptive tolerance, and the convergence of the value of the adaptive tolerance is reached in a low number of adjustments.