# Viscothermal models for wind musical instruments

1 Magique 3D - Advanced 3D Numerical Modeling in Geophysics
LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau], Inria Bordeaux - Sud-Ouest
Abstract : This work focuses on thermal and viscous effects on linear wave propagation inside apipe. It aims at understanding the ground on which are built many dissipative propagating wavemodels found in the musical acoustics literature, in order to quantify, as much as possible, theunderlying assumptions and model errors which are performed. The Navier-Stokes (NS) equations,which are nonlinear and expressed in the 3 dimensions of space, are the starting point of allmodels. Thermoviscous (or viscothermal) equations are derived from NS equations mainly afterlinearization and assumptions on the gas state equation. Analytical or numerical solutions to theseequations can be proposed, after modifying more of less the original system. These derived modelsare summed-up in a global sketch of the underlying hypotheses. What is observed is that the thermal and viscous effects are mainly confined near the boundaries of the pipe, in regions called "boundary layers", whose lengths depend on the physical coefficients and the harmonic regime.These thermal and viscous effects can therefore be neglected far from the boundaries (where astandard 3D Helmholtz wave equation holds). These procedures can lead to 3D models, describing the propagation of the pressure field in all the domain or only a part of it, or 1D models, describing the propagation of the mean pressure across a pipe section. Some other 1D models are obtained from derived 3D models, and describe the propagation of the pressure near the boundary layer.The time dependancy of these derived models can be very intricate because the model derivation is done in the harmonic regime, and nonlocal operators can arise. An additional modelling step can lead to local time-domain 1D models. An assessment of some of these 3D and 1D models is proposed, aiming at a quantitative estimation of the model errors with respect to their domains of validity, and at a comparison between some models for simple geometries.
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Reports

https://hal.inria.fr/hal-02917351
Contributor : Juliette Chabassier <>
Submitted on : Wednesday, August 19, 2020 - 9:53:53 AM
Last modification on : Friday, January 15, 2021 - 9:20:33 AM
Long-term archiving on: : Monday, November 30, 2020 - 9:33:23 PM

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RR-9356.pdf
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• HAL Id : hal-02917351, version 1

### Citation

Alexis Thibault, Juliette Chabassier. Viscothermal models for wind musical instruments. [Research Report] RR-9356, Inria Bordeaux Sud-Ouest. 2020. ⟨hal-02917351⟩

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