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Streszczenie:
This work presents benchmark examples related to the modelling of sound absorbing porous media with rigid frame based on the periodic geometry of their microstructures. To this end, rigorous mathematical derivations are recalled to provide all necessary equations, useful relations, and formulae for the so-called direct multi-scale computations, as well as for the hybrid multi-scale calculations based on the numerically determined transport parameters of porous materials. The results of such direct and hybrid multi-scale calculations are not only cross verified, but also confirmed by direct numerical simulations based on the linearised Navier-Stokes-Fourier equations. In addition, relevant theoretical and numerical issues are discussed, and some practical hints are given.

Słowa kluczowe:
porous media, periodic microstructure, wave propagation, sound absorption

Streszczenie:
Samples with periodic microstructures, designed for good sound absorption, have been manufactured by 3D printing. Typically, however, the acoustical properties of the resulting samples differ from those predicted. Two causes of the discrepancies are (1) inaccuracies related to the 3D-printing resolution and (2) imperfections resulting from micro-fibres, micro-pores, and pore surface roughness, created during manufacture. Discrepancies due to the first cause can be addressed, post hoc, by updating the idealised periodic geometric model used for creating the codes for fabrication on the basis of a survey using a scanning microscope, or through computerised micro-tomography scans. Reducing the discrepancies due to the second cause requires a relatively significant further modelling effort. Another cause for small discrepancies is when two layers of the same periodic porous material and thickness differ only by a relative shift of the internal geometry of the periodic Representative Volume Element (RVE). This causes the absorption peaks to be shifted in frequency. A modelling procedure is proposed to take this into account.

Słowa kluczowe:
Sound absorption, Periodic porous media, Additive manufacturing

Streszczenie:
Numerical simulation of absorption of sound in porous media is an important part of the design of the treatments for the environmental noise reduction. In the porous media, the mechanical energy carried by sound is dissipated by thermo-viscous interactions with the solid surface of the media frame, which usually has complicated geometry at the microscopic (sub-millimetre) scale. In order to be able to absorb the acoustic energy at the low frequencies of interest, a layer of porous material must be rather thick (at the order of centimetres). This is why direct numerical simulation (DNS) of the sound absorption in porous media is a rather computationally challenging task because small geometrical details must be properly resolved in a large computational domain. In order to avoid these difficulties, simplified semi-phenomenological models introducing so called effective fluid have been proposed. For example, the Johnson-Champoux-Allard-Pride-Lafarge (JCAPL) model is based on eight parameters which can be measured or calculated based on the media micro-structural geometry. Within this work, we compare the numerical results obtained by the 3D DNS with the prediction of the JCAPL model in case of several porous media represented by closely-packed spheres. The DNS calculations are performed using the linearised Navier-Stokes equations for layers of spheres of different thicknesses, the parameters for the JCAPL model are calculated subsequently using Laplace, Poisson, and Stokes flow analyses on a representative volume element of the media. Very good agreement between the results has been found.