Asymptotic Analysis and Boundary Layers

Cousteix, J.; Mauss, Jacques

doi:10.1007/978-3-540-46489-1

Cited by 62 publications

(95 citation statements)

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“…These matching rules can be stated in numerous manners (cf., e.g., [6,13,5,17]). We have used the following approaches, previously introduced by two of the authors [3,4], which seemed to us to be the most appropriate in this context.…”

Section: The Matched Asymptotic Expansionsmentioning

confidence: 99%

“…Once the existence of the inner and outer expansions is established, a uniformly valid approximation of the reflected and transmitted waves can be built (cf., e.g., [4,5,16]) and error estimates proved, thus completing the justification process. This part, quite straightforward but with a lot of technicalities, is not considered here.…”

Section: Determination Of the Second-order Outer Expansionmentioning

confidence: 99%

“…The two-scale expansions. We use the method of two-scale matched asymptotic expansions [6,13,5,17] below to build an expansion which enables us to get a sharp approximation of u δ outside "asymptotic" neighborhoods of each perforation and another one valid inside them. In the terminology of this method, these expansions are respectively called the outer and the inner expansions.…”

Section: The Matched Asymptotic Expansionsmentioning

confidence: 99%

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

Bendali¹,

Fares²,

Piot³

et al. 2013

SIAM J. Appl. Math.

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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

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Section: The Matched Asymptotic Expansionsmentioning

confidence: 99%

Section: Determination Of the Second-order Outer Expansionmentioning

confidence: 99%

Section: The Matched Asymptotic Expansionsmentioning

confidence: 99%

See 1 more Smart Citation

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

Bendali¹,

Fares²,

Piot³

et al. 2013

SIAM J. Appl. Math.

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show abstract

“…We are thus led to use the method of matched asymptotic expansions to construct an expansion for the field outside the immediate proximity of the balls and another one inside a boundary layer enclosing each of them. Relevant introductions to the method of matched asymptotic expansions can be found, for instance in, [8,11,7]. According to the terminology in use for this method, any issue concerning the expansion outside the proximity of the scatterers and inside the boundary layers will be referred to as outer and inner, respectively.…”

Section: Matched Asymptotic Expansionsmentioning

confidence: 99%

“…Consequently, such a method does not directly yield the asymptotic expansion of the scattered wave. To obtain this expansion, we use the method of matched asymptotic expansions (cf., e.g., [8,9,11,7,4]) instead. The main advantage of this procedure is that it yields a convenient way to describe both the field inside a boundary layer enclosing each sphere and the overall behavior of the scattered field.…”

Section: Introductionmentioning

confidence: 96%

Scattering of a scalar time-harmonic wave by N small spheres by the method of matched asymptotic expansions

Bendali

Cocquet

Tordeux

2012

Numer. Analys. Appl.

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Real Analytic Local Well‐Posedness for the Triple Deck

Iyer

Vicol

2020

Comm Pure Appl Math

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The Triple Deck model is a classical high order boundary layer model that has been proposed to describe flow regimes where the Prandtl theory is expected to fail. At first sight the model appears to lose two derivatives through the pressure-displacement relation which links pressure to the tangential slip. In order to overcome this, we split the Triple Deck system into two coupled equations: a Prandtl type system on H and a Benjamin-Ono type equation on R. This splitting enables us to extract a crucial leading order cancellation at the top of the lower deck. We develop a functional framework to subsequently extend this cancellation into the interior of the lower deck, which enables us to prove the local well-posedness of the model in tangentially real analytic spaces.

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Asymptotic Analysis and Boundary Layers

Cited by 62 publications

References 0 publications

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

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

Scattering of a scalar time-harmonic wave by N small spheres by the method of matched asymptotic expansions

Real Analytic Local Well‐Posedness for the Triple Deck

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