Multimodal nonlinear acoustics in two- and three-dimensional curved ducts

We develop a weakly nonlinear model of duct acoustics in two and three dimensions (without flow). The work extends the previous work of McTavish & Brambley (2019 J. Fluid Mech. , vol. 875 , pp. 411–447) to three dimensions and significantly improves the numerical efficiency. The model allows for general curvature and width variation in two-dimensional ducts, and general curvature and torsion with radial width variation in three-dimensional ducts. The equations of gas dynamics are perturbed and expanded to second order, allowing for wave steepening and the formation of weak shocks. The resulting equations are then expanded temporally in a Fourier series and spatially in terms of straight-duct modes, and a multi-modal method is applied, resulting in an infinite set of coupled ordinary differential equations for the modal coefficients. A linear matrix admittance and its weakly nonlinear generalisation to a tensor convolution are first solved throughout the duct, and then used to solve for the acoustic pressures and velocities. The admittance is useful in its own right, as it encodes the acoustic and weakly nonlinear properties of the duct independently from the specific wave source used. After validation, a number of numerical examples are presented that compare two- and three-dimensional results, the effects of torsion, curvature and width variation, acoustic leakage due to curvature and nonlinearity and the variation in effective duct length of a curved duct due to varying the acoustic amplitude. The model has potential future applications to sound in brass instruments. Matlab source code is provided in the supplementary material.