On the homogenization of the acoustic wave propagation in perforated ducts of finite length for an inviscid and a viscous model

Abstract

The direct numerical simulation of the acoustic wave propagation in multiperforated absorbers with hundreds or thousands of tiny openings would result in a huge number of basis functions to resolve the microstructure. One is, however, primarily interested in effective and so homogenized transmission and absorption properties and how they are influenced by microstructure and its endpoints. For this, we introduce the surface homogenization that asymptotically decomposes the solution in a macroscopic part, a boundary layer corrector close to the interface and a near-field part close to its ends. The effective transmission and absorption properties are expressed by transmission conditions for the macroscopic solution on an infinitely thin interface and corner conditions at its endpoints to ensure the correct singular behaviour, which are intrinsic to the microstructure. We study and give details on the computation of the effective parameters for an inviscid and a viscous model and show their dependence on geometrical properties of the microstructure for the example of Helmholtz equation. Numerical experiments indicate that with the obtained macroscopic solution representation one can achieve an high accuracy for low and high porosities as well as for viscous boundary conditions while using only a small number of basis functions.

On impedance conditions for circular multiperforated acoustic liners

“…This blow up of the coefficient D δ ∞ as δ → 0 in accordance with its numerical computations based on an asymptotic analysis of (3) with only two scales [8], where the hole size is considered not as a scale but as a parameter.…”

“…To see the analogy it suffices to consider time harmonic fields varying like exp(−ıωt), the volume flux Q(t) through the aperture counted positively along the direction of the e r axis to be the same as the volume flux through the surface Γ + (S) (respectively Γ − (S)), counted positively (resp. negatively) along the direction of the normal vector n, and to compare (1) and (8). Note, that the normal component of the near field velocity profile v decays like 1/S 2 towards infinity and combines different behaviour close to and away from the wall (see Fig.…”

Surface homogenization of an array of Helmholtz resonators for a viscoacoustic model using two-scale convergence

“…Considering a period δ that is small in comparison to the wave-length and even smaller diameter of the orifices, scaled as δ 2 , and small viscosity that is scaled like δ 4 we obtain a nontrivial limit for δ → 0, in difference to an homogenization with only two geometric scales, cf. [8,51]. In this way the dominating effects are considered on each geometric scale, that are viscous effects and incompressible acoustic velocity around each hole, only incompressibility in an intermediate zone above and below the perforated plate, one-dimension wave-propagation inside the resonance chamber and pure acoustics wave-propagation away from the perforated plate.…”

“…The effective Rayleigh conductivity depends on the geometrical parameters, especially, size and shape of the necks of the Helmholtz resonators and the distance between two resonators, as well as the physical parameters, especially the acoustic viscosities and the excitation frequency. Asymptotic homogenization for periodic transmission problems were performed for the Stokes equation with three scales [44], with two scales for the Helmholtz equation [8], using the periodic unfolding method [31] and the method of matched asymptotic expansion [14,12], also with impedance boundary conditions in the holes [51]. Asymptotic homogenization for locally periodic transmission problems, where microstructures has finite size, and that takes the singular behaviour at the end of the microstructure into account, was derived and justified for the Laplace equation [34,15] and the Helmholtz equation [50].…”

Homogenization of thin-structured surfaces for acoustics in the presence of a two-dimensional low Mach potential flow

“…In [13], not limited by the geometry, a two-scale homogenization method based on a matched asymptotic expansion technique was used, leading to a boundary condition obtained on an equivalent flat wall. Such surface homogenization and more generally interface homogenizations based on matched asymptotic expansion techniques have been widely used, in acoustics [14][15][16][17] but also in different wave contexts like elastic waves [18] or electromagnetic waves [19][20][21][22].…”