Mathematical Justification of the Rayleigh Conductivity Model for Perforated Plates in Acoustics

Abstract

This paper is devoted to the mathematical justification of the usual models predicting the effective reflection and transmission of an acoustic wave by a low porosity multiperforated plate. Some previous intuitive approximations require that the wavelength be large compared with the spacing separating two neighboring apertures. In particular, we show that this basic assumption is not mandatory. Actually, it is enough to assume that this distance is less than a half-wavelength. The main tools used are the method of matched asymptotic expansions and lattice sums for the Helmholtz equations. Some numerical experiments illustrate the theoretical derivations.1. Introduction. Perforated plates are widely used in engineering systems, either as resistive sheets for acoustic liners [24] or for cooling purposes at the walls of the combustion chambers of jet engines [19]. As these plates generally contain tens of thousands of holes for realistic geometries, the direct determination of the effect of the perforations is out of reach in numerical simulations. As a result, the effective acoustic behavior of such plates is usually reproduced by designing suitable semi-analytic models.Most of these models are based on the calculation of the Rayleigh conductivity of a hole, which relates the net fluctuating volume flux through the perforation to the fluctuating pressure drop across it (cf., e.g., [32,12]). Such an approach has several advantages. The most valuable of them is the possibility of coupling models taking into account the absorption of acoustic energy by viscosity at the orifices with the usual equations governing the propagation of acoustic waves elsewhere. The most frequently used model relies on the expression for the conductivity of a single circular hole in an infinitely thin wall derived by Howe [10]. In this model, a bias flow crosses the orifice. The limiting case, when the velocity of the bias flow goes to zero, is reduced to the very classical approach, earlier derived by Rayleigh [32]. In [35], one can find a clear derivation of the Rayleigh conductivity model for circular perforations in an infinitely thin plate, although formal and implicit as regards the method of matched asymptotic expansion. Howe's model was later extended to a hole in a thick plate [11,15]. These models are widely used in the literature for plates of low porosity

Effective conditions for the reflection of an acoustic wave by low-porosity perforated plates

“…In this section, we focus on a method for determining the asymptotic expansion of the reflection and transmission coefficients of the aforementioned low-porosity perforated plate up to order 2 relative to the small parameter characterizing the ratio of a characteristic size d of the perforation to the spacing L between two neighbouring perforations. This expansion has been recently given in Bendali et al (2013). However, contrary to the approach in this reference, which was based on complex two-scale matched asymptotic expansions and the consideration of a grating of multipoles, the present one deals with these coefficients in a direct way without resorting to the expansion of the whole wave.…”

“…The main mathematical tools are asymptotic expansions, integral equations, and the lattice sum theory for the Helmholtz equation (Linton 2010). We end this section by briefly recalling results from Bendali et al (2013) on the derivation of effective compliances for the plate from the asymptotic expansion of the reflection and transmission coefficients.…”

“…The reflection and transmission properties of such panels perforated with a periodic array of holes can be modeled directly, as was done by Leppington & Levine (1973) by means of an integral equation or later by the method of matched asymptotic expansions (see Bendali et al (2013) and Leppington (1990) for the case of elastic plates). The acoustic effect of perforated plates is usually addressed by studying the acoustic behaviour of a single aperture in an infinite wall and by designing a homogeneous model for the whole plate, mainly by means of an averaging procedure.…”

“…In these papers, the asymptotic solution was sought for the long wavelength limit and under the low-porosity assumption, i.e., the size of each perforation is assumed to be small in comparison to the distance between two neighbouring apertures of the periodic array, viewed in turn as being much smaller than the wavelength of the impinging sound waves. In Bendali et al (2013), it is the whole wave which is expanded by means of the method of matched asymptotic expansions. As a result, the procedure for determining the two top-order terms of the asymptotic expansions of the reflection and transmission coefficients relative to the small parameter, characterizing the ratio of a characteristic size of the perforation to the spacing between two successive perforations, is particularly complex.…”

Impermeability Through a Perforated Domain for the Incompressible two dimensional Euler Equations

“…Most papers have focused on the Laplace equation [8], on Stokes and Navier-Stokes flows [1,29,32,34]. Recently, more attention was given to the homogenization of other fluid models such as the compressible Navier-Stokes system [11,27] and the acoustic system [4,12].…”