Modal approaches for modelling nonlinear acoustics in 2D axisymmetric domains: application to brass instruments

The modelling approach presented here aims at developing accurate, yet computationally efficient, numerical models of brass instrument resonators including nonlinear propagation, viscothermal losses and 2D radiation effects.Firstly, we propose the use of the Blackstock equation to model the nonlinear acoustic propagation inside the resonator, a more appropriate choice when dealing with nonlinear standing wave patterns, compared to the commonly used Burger's equation.The initial step of the approach consists in obtaining a complex modal basis from a 2D-axisymmetric finite element model of the linearized equations.Here, we include a bounded domain outside the resonator with a nonreflecting boundary condition as well as the effect of viscothermal losses at the interior walls.The nonlinear Blackstock equation is then projected onto the resulting 2D complex modal basis, leading to a compact set of nonlinear ODEs.This leads to exploitable reduced formulations adapted to quick temporal simulations, bifurcation analysis or parametric studies, retaining nevertheless the accuracy of 2D models.The explicit account of the exterior acoustic field also allows for the calculation of radiated sound pressures as well as directivity patterns.Experimental validation and illustrative numerical results are presented for a simplified trumpet geometry in both linear and nonlinear scenarios.