Numerical modeling of nonlinear acoustic waves in a tube with an array of Helmholtz resonators

Bruno Lombard (1) , Jean-François Mercier (2)

1 O&I - Ondes et Imagerie
2 POEMS - Propagation des Ondes : Étude Mathématique et Simulation

Wave propagation in a 1-D guide with an array of Helmholtz resonators is studied numerically, considering large amplitude waves and viscous boundary layers. The model consists in two coupled equations: a nonlinear PDE of nonlinear acoustics, and a linear ODE describing the oscillations in the Helmholtz resonators. The dissipative effects in the tube and in the throats of the resonators are modeled by fractional derivatives. Based on a diffusive representation, the convolution kernels are replaced by a finite number of memory variables that satisfy local ordinary differential equations. An optimization procedure provides an efficient diffusive representation. A splitting strategy is then applied to the evolution equations: the propagative part is solved by a standard TVD scheme for hyperbolic equations, whereas the diffusive part is solved exactly. This approach is validated by comparisons with exact solutions. The properties of the full nonlinear solutions are investigated numerically. In particular, existence of acoustic solitary waves is confirmed.

Mots clés: nonlinear acoustics; solitons; Burgers equation; fractional derivatives; diffusive representation; time splitting; shock-capturing schemes

Domaines: Acoustique [physics.class-ph]; Acoustique [physics.class-ph]; Analyse numérique [math.NA]

Bruno Lombard, Jean-François Mercier. Numerical modeling of nonlinear acoustic waves in a tube with an array of Helmholtz resonators. 2013. ⟨hal-00843609v1⟩