Nonlinear acoustics in brass instruments using 2D complex modal bases

The modelling approach presented here aims at developing accurate, yet computationally efficient, numerical models of brass instrument resonators including nonlinear propagation, viscothermal losses and radiation effects in 2D axisymmetric domains. The central idea is to bridge the refined accuracy of 2D models with the efficiency of reduced-order modal approaches. To model the nonlinear acoustic propagation inside the resonator, we propose the use of the Blackstock equation, a more appropriate choice when dealing with nonlinear standing wave motion, compared to the more commonly used Burgers equation. The first step of the approach consists in obtaining a complex modal basis from a 2D axisymmetric finite element model of the linearized equations. Here, a bounded domain outside the resonator, with a nonreflecting boundary condition, is included to explicitly account for 2D radiation effects. Moreover, the effects of the viscothermal boundary layers at the interior walls are modelled efficiently through an impedance boundary condition. The Blackstock equation is then projected onto the resulting 2D complex modal basis, leading to a compact set of nonlinear ordinary differential equations (ODEs) that can be truncated at any suitable number of terms. This leads to exploitable reduced formulations, adapted to quick temporal simulations, bifurcation analysis or parametric studies, retaining nevertheless the accuracy provided by 2D axisymmetric models. Additionally, the explicit account of the exterior acoustic field allows for the calculation of radiated sound pressures as well as directivity patterns, contrary to typical 1D models. Experimental validation and illustrative numerical results are presented for a simplified trumpet geometry in both linear and nonlinear scenarios, highlighting the benefits of the proposed approach.