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Conference Papers Year : 2021

A low reduced frequency model for viscothermal wave propagation in conical tubes of arbitrary cross-section

Abstract

Accurate computation of acoustic propagation in conical wind instruments, such as the bassoon or oboe, must include dissipation and dispersion due to viscothermal coupling. These boundary layer effects may be taken into account in the volume of the 3D domain, on the walls as effective boundary conditions, or using complex coefficients in 1D reduced models. Whereas the expressions of these coefficients are commonly derived in the harmonic domain for cylindrical tubes, leading for instance to the Zwikker-Kosten model, the current work aims to investigate their variation with respect to conicity, and to analyse stability and passivity of corresponding immittances in the Laplace domain. To this end, a simplification of the linearized Navier-Stokes equations in straight (respectively conical) tubes with arbitrary cross-sectional shapes is considered, where surfaces of constant pressure are assumed to be planar (respectively spherical). Under the assumption that the wavelength is much larger than both the tube radius and the boundary layer thickness, the unknown pressure and flow are shown to be solutions of a pair of coupled 1D differential equations, formulated as telegraphist's equations involving complex coefficients. In the classical case of cylinders with circular cross-section, the formulas are shown to coincide with the Zwikker-Kosten theory. The expression of coefficients for cones with circular cross-section is derived. It involves complex-order spherical harmonics, instead of Bessel functions for cylinders, and makes the hydraulic radius appear as the relevant geometrical parameter. A practical and accurate approximation of the new model for conical shaped instruments is to use a modified Zwikker-Kosten model with spherical wavefronts, where the geometrical radius is replaced with the hydraulic radius. Cone impedances are computed numerically with finite elements methods and compared for several 1D and 3D models.
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Dates and versions

hal-03332223 , version 1 (27-09-2022)

Identifiers

  • HAL Id : hal-03332223 , version 1

Cite

Alexis Thibault, Juliette Chabassier, Henri Boutin, Thomas Hélie. A low reduced frequency model for viscothermal wave propagation in conical tubes of arbitrary cross-section. ICSV27 - 27th International Congress on Sound and Vibration, Jul 2021, Prague, Czech Republic. ⟨hal-03332223⟩
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