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The Influence of a Pressure Wavepacket's Characteristics on Its The influence of a pressure wavepacket’s characteristics on its acoustic radiation R. Serré, Jean-Christophe Robinet, F. Margnat To cite this version: R. Serré, Jean-Christophe Robinet, F. Margnat. The influence of a pressure wavepacket’s character- istics on its acoustic radiation. Journal of the Acoustical Society of America, Acoustical Society of America, 2015, 137 (6), pp.3178-3189. 10.1121/1.4921031. hal-02569482 HAL Id: hal-02569482 https://hal.archives-ouvertes.fr/hal-02569482 Submitted on 11 May 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The influence of a pressure wavepacket’s characteristics on its acoustic radiation R. Serre and J.-C. Robinet Arts et Metiers ParisTech, DynFluid, 151 Boulevard de l’Hopital, 75013 Paris, France a) F. Margnat Institut PPRIME, Department of Fluid Flow, Heat Transfer and Combustion, Universite de Poitiers - ENSMA - CNRS, Building B17 - 6 rue Marcel Dore, TSA 41105, 86073 Poitiers Cedex 9, France Noise generation by flows is modeled using a pressure wavepacket to excite the acoustic medium via a boundary condition of the homogeneous wave equation. The pressure wavepacket is a generic representation of the flow unsteadiness, and is characterized by a space envelope of pseudo- Gaussian shape and by a subsonic phase velocity. The space modulation yields energy in the super- sonic range of the wavenumber spectrum, which is directly responsible for sound radiation and di- rectivity. The influence of the envelope’s shape on the noise emission is studied analytically and numerically, using an acoustic efficiency defined as the ratio of the acoustic power generated by the wavepacket to that involved in the modeled flow. The methodology is also extended to the case of acoustic propagation in a uniformly moving medium, broadening possibilities toward practical flows where organized structures play a major role, such as co-flow around cruising jet, cavity, and turbulent boundary layer flows. The results of the acoustic efficiency show significant sound pres- sure levels, especially for asymmetric wavepackets radiating in a moving medium. I. INTRODUCTION definition of a source term as the acoustic excitation; that is solving an inhomogeneous wave equation. The Lighthill for- The goal of the present study is to estimate an acoustic malism is the natural candidate to this aim. It has been car- efficiency of unsteady flows, defined as the rate of energy ried out by Obrist8 who modeled the first component of the that aerodynamic fluctuations are able to leak into acoustic Lighthill tensor as a wavepacket, and investigated the role of waves. That problem is addressed here from the acoustical its phase velocity and its spatial distribution in a multi- point of view, that is, searching for which kind of excitation dimensional space, then extending to that of the group veloc- the acoustic medium responds the most (or the least) effi- ity.9 That author emphasized the predominant effects of ciently. In order to model flow induced noise, a generic exci- those characteristics on directivity patterns. The same for- tation may be a wave whose phase speed is subsonic and malism has been used through analytical and experimental whose amplitude is spatially modulated.1,2 This accounts for work by Papamoschou10 and Cavalieri et al.11 For jittering a localized shear region embedded in the acoustic medium, wavepackets in a middle subsonic flow, the temporal fluctua- as one can find, e.g., within a spatially evolving mixing- tions of the envelope highlighted efficient conditions for layer: Kelvin-Helmholtz vortices, due to a harmonic forcing sound radiation. The second technique is to excite the wave inflow, are convectively amplified before the pairing phe- operator through the boundary conditions of the homogeneous nomenon occurs. The hydrodynamic pressure fluctuation wave equation as used by Avital and Sandham12 (see also associated with the vortices then has the form of such a Refs. 13–15, for instance). In particular cases, an analytical wavepacket along the mixing layer.3,4 Such radiation by solution might be provided, as Crighton and Huerre16 did to amplified hydrodynamic fluctuations also occurs at the trail- suggest a theoretical explanation for the directivity of forced ing edge of a sharp-edged flat plate.5 A solution of this prob- jets that Laufer and Yen17 observed in their experiments. The lem is critical to the efficient design of control strategies, same track was followed by Fleury et al.18 on a cylindrical because the latter may be different whether the acoustic Kirchhoff surface in the context of round jet noise. power output is due to a high amount of energy brought by The acoustic energy radiated by a pressure distribution the hydrodynamic flow or a high efficiency through the con- is triggered by its modes in the wavenumber spectrum version process into acoustics. lower than the acoustic wavenumber (supersonic phase The acoustic response to generic pressure distributions speed9,15,19,20). The localized spatial envelope yields an is driven by a Cauchy problem whose direct numerical reso- amplitude modulation in the wavenumber spectrum, instead lution as a partial differential equation is difficult.6,7 Two of a single pic at the hydrodynamic wavenumber in the other techniques are available. The first one consists in the case of a wave without space modulation. Consequently, even for subsonic phase speed, there is energy in the super- sonic range of the wavenumber spectrum. This is usually a)Electronic mail: fl[email protected] referred to as the supersonic tail of the wavepacket and can be considered as the purely radiating component of it. The @2 @2 1 @2 p^ x; t þ p^ x; t À p^ x; t ¼ 0 remaining part of the wavenumber spectrum of the pressure 2 ðÞ 2 ðÞ 2 2 ðÞ @x1 @x2 c1 @t distribution has mathematically no contribution to the þ acoustic field in the Kirchhoff formalism. It may be con- with x1 ʦ R and x2 ʦ R . The Fourier transform in time, ceptually related to the silent base flow introduced by defined as follows, is applied, yielding the homogeneous Sinayoko and Agarwal,21 whose physical existence and Helmholtz equation: meaning has not yet been clearly established, however. On 8 ð > þ1 the other hand, which proportion does that acoustic energy > > f ðx; xÞ¼ f^ðx; tÞeþixtdt; represent with respect to the energy brought by the hydro- < 1ð dynamic wave needs further documentation, as well as the > þ1 > 1 dependence of this proportion on the wavepackets :> f^ðx; tÞ¼ f ðx; xÞeÀixtdx; 2p characteristics. 1 The present paper follows the guideline set by Sandham et al.14 through numerical integration of the homogeneous where i is the imaginary unit and x is the angular frequency. Helmholtz equation in the Kirchhoff formalism, introducing For one particular mode, one obtains the acoustic excitation as a boundary condition. Salient fea- @2 @2 tures of wavepackets are their shape, designed by the enve- pðÞx þ pðÞx þ k2pðÞx ¼ 0; (1) @x2 @x2 a lope, and the wavenumber of the traveling waves within the 1 2 latter. The purpose of this study is to characterize the acous- with k ¼ x=c . The excitation, assumed to be harmonic in tic response of the medium to such generic excitation, a 1 frequency, is introduced through the boundary condition (x through a parametric investigation on the envelope function, 2 ¼ 0), in the following general form: assumed static. The effect of both the hydrodynamic and the acoustic compactness of the wavepacket is investigated, x pxðÞ1; 0 ¼ P0AxðÞ1 exp i x1 ; (2) varying the envelope length and the phase Mach number of Uc the traveling wave, respectively. The effect of an asymmetri- cal envelope is specifically studied. The acoustic response is where A is the envelope function, Uc is the phase velocity of described by the acoustic efficiency of the wavepacket and the convected wave, and P0 is the pressure amplitude. The how the acoustic intensity is distributed on the directivity latter is taken as unity and will be omitted in the following, range. Propagation in an acoustic medium at rest is first con- unless otherwise specified. The problem is sketched up in sidered. Second, the analysis is extended when a uniform Fig. 1. flow is present in the acoustic medium, using a convected We resolve the Helmholtz Eq. (1) in the Kirchhoff for- Green function. It thus broadens possibilities toward flows malism24 which yields the acoustic pressure according to the where organized, unsteady motions may be identified as initial distribution dominant aeroacoustic sources such as cavities22 and turbu- ð lent boundary layers.23 @pðÞy @GðÞxjy pðÞx ¼ GðÞxjy À pðÞy dRðÞy ; The paper is organized as follow: Section II explicits the RðÞy @n @n theoretical background with the mathematical conventions (3) hereby adopted for the resolution of the wave equation; the Kirchhoff formalism is introduced in a static, propagation R(y) is the control surface and n its outward pointing nor- medium and a formulation for a distant observer is given. mal. The two-dimensional (2D) free-space Green function Analytical developments yielded prediction for high effi- associated with the above Fourier transform convention is ciency conditions; the numerical tool is presented in Sec. III, and its range of validity and interest is exhibited by compari- son with the results given by the analytical formulas.
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