×

zbMATH — the first resource for mathematics

Acoustic scattering at a hard-soft lining transition in a flow duct. (English) Zbl 1198.76136
Author’s abstract: An explicit Wiener-Hopf solution is derived to describe the scattering of sound at a hard-soft wall impedance transition at \(x = 0\), say, in a circular duct with uniform mean flow of Mach number \(M\). A mode, incident from the upstream hard section, scatters at \(x = 0\) into a series of reflected modes and a series of transmitted modes. Of particular interest is the role of a possible instability along the lined wall in combination with the edge singularity. If one of the “upstream” running modes is to be interpreted as a downstream-running instability, an extra degree of freedom in the Wiener-Hopf analysis occurs that can be resolved by application of some form of Kutta condition at \(x = 0\), for example a more stringent edge condition where wall streamline deflection \(h={\mathcal{O}}(x^{3/2})\) at the downstream side. In general, the effect of this Kutta condition is significant, but it is particularly large for the plane wave at low frequencies and should therefore be easily measurable. For small Helmholtz numbers, the reflection coefficient modulus \(| R_{001}|\) tends to \((1 + M)/(1 - M)\) without and to 1 with Kutta condition, while the end correction tends to \(\infty\) without and to a finite value with Kutta condition. This is exactly the same behaviour as found for reflection at a pipe exit with flow, irrespective if this is uniform or jet flow. Although the presence of the instability in the model is hardly a question anymore since it has been confirmed numerically, a proper mathematical causality analysis is still not totally watertight. Therefore, the limit of a vortex sheet, separating zero flow from mean flow, approaching the wall has been explored. Indeed, this confirms that the Helmholtz unstable mode of the free vortex sheet transforms into the suspected mode and remains unstable. As the lined-wall vortex-sheet model predicts unstable behaviour for which experimental evidence is at best rare and indirect, the question may be raised if this model is indeed a consistent simplification of reality, doing justice to the double limit of small perturbations and a thin boundary layer. Numerical time-domain methods suffer from this instability, and it is very important to decide whether the instability is at least physically genuine. Experiments based on the present problem may provide a handle to resolve this stubborn question.

MSC:
76Q05 Hydro- and aero-acoustics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Crighton DG and Pedley TJ (1999). Michael James Lighthill (1924–1998). Notices AMS 46(1): 1226–1229 · Zbl 1194.01083
[2] Pedley TJ (2001). James Lighthill and his contributions to fluid mechanics. Annu Rev Fluid Mech 33: 1–41 · Zbl 1032.01531 · doi:10.1146/annurev.fluid.33.1.1
[3] Lighthill MJ (1952). On sound generated aerodynamically, I. General theory. Proc R Soc Lond A 211: 564–587 · Zbl 0049.25905 · doi:10.1098/rspa.1952.0060
[4] Crighton DG (1981). Acoustics as a branch of fluid mechanics. J Fluid Mech 106: 261–298 · Zbl 0472.76002 · doi:10.1017/S0022112081001602
[5] Lighthill MJ (1954). On sound generated aerodynamically II. Turbulence as a source of sound. Proc R Soc Lond A 222: 1–32 · Zbl 0055.19109
[6] Lighthill MJ (1962). Sound generated aerodynamically, the Bakerian Lecture 1961. Proc R Soc Lond A 267: 147–182 · Zbl 0104.43402 · doi:10.1098/rspa.1962.0090
[7] Lighthill MJ (1993). A general introduction to aeroacoustics and atmospheric sound. In: Hardin, JC and Hussaini, MY (eds) Computational Aeroacoustics, pp. Springer-Verlag, New York
[8] Stein RF (1967). Generation of acoustic and gravity waves by turbulence in an isothermal stratified atmosphere. Solar Phys 2: 385–432 · doi:10.1007/BF00146490
[9] Crighton DG (1975). Basic principles of aerodynamic noise generation. Prog Aerosp Sci 16: 13–96 · doi:10.1016/0376-0421(75)90010-X
[10] Howe MS (2001). Vorticity and the theory of aerodynamic sound. J Eng Math 41(4): 367–400 · Zbl 1013.76076 · doi:10.1023/A:1012362222597
[11] Crighton DG (1988) Aeronautical acoustics: mathematics applied to a major industrial problem. In: McKenna J, Temam R (eds) Proceedings of the first international conference on industrial and applied mathematics ICIAM’87. SIAM, Philadelphia, pp 75–89
[12] Smith MJT (1989) Aircraft noise. Cambridge University Press
[13] Lighthill MJ (1972). The propagation of sound through moving fluids, the fourth annual fairey lecture. J Sound Vibration 24: 472–492 · Zbl 0246.76087 · doi:10.1016/0022-460X(72)90718-3
[14] Swinbanks MA (1975). The sound field generated by a source distribution in a long duct carrying sheared flow. J Sound Vibration 40(1): 51–76 · doi:10.1016/S0022-460X(75)80230-6
[15] Lighthill MJ (1965). Group velocity. J Inst Math Appl 1: 1–28 · Zbl 0216.52201 · doi:10.1093/imamat/1.1.1
[16] Lighthill MJ (1960). Studies on magneto-hydrodynamic waves and other anisotropic wave motions. Phil Trans R Soc Lond A 252: 397–430 · Zbl 0097.20806 · doi:10.1098/rsta.1960.0010
[17] Lighthill MJ (1978) Waves in fluids. Cambridge University Press · Zbl 0375.76001
[18] Rienstra SW (1999). Sound transmission in slowly varying circular and annular lined ducts with flow. J Fluid Mech 380: 279–296 · Zbl 0946.76088 · doi:10.1017/S0022112098003607
[19] Rienstra SW (2003). Sound propagation in slowly varying lined flow ducts of arbitrary cross-section. J Fluid Mech 495: 157–173 · Zbl 1052.76063 · doi:10.1017/S0022112003006050
[20] Rademaker ER (1990) Experimental validation of a lined-duct acoustics model including flow. Presented at ASME conference on duct acoustics, Dallas, TX, Nov. 1990
[21] Ingard KU (1959). Influence of fluid motion past a plane boundary on sound reflection, absorption and transmission. J Acoust Soc Am 31(7): 1035–1036 · doi:10.1121/1.1907805
[22] Myers MK (1980). On the acoustic boundary condition in the presence of flow. J Sound Vibration 71(3): 429–434 · Zbl 0448.76065 · doi:10.1016/0022-460X(80)90424-1
[23] Eversman W and Beckemeyer RJ (1972). Transmission of sound in ducts with thin shear layers–convergence to the uniform flow case. J Acoust Soc Am 52(1): 216–220 · Zbl 0238.76020 · doi:10.1121/1.1913082
[24] Rienstra SW (2003). A classification of duct modes based on surface waves. Wave Motion 37(2): 119–135 · Zbl 1163.74431 · doi:10.1016/S0165-2125(02)00052-5
[25] Tester BJ (1973). The propagation and attenuation of sound in ducts containing uniform or ’Plug’ flow. J Sound Vibration 28(2): 151–203 · Zbl 0258.76057 · doi:10.1016/S0022-460X(73)80102-6
[26] Bers A, Briggs RJ (1963) MIT Research Laboratory of Electronics Report No. 71 (unpublished)
[27] Briggs RJ (1964) Electron-stream interaction with plasmas. Monograph no. 29, MIT Press, Cambridge Massachusetts
[28] Bers A (1983) Space-time evolution of plasma instabilities–absolute and convective. In: Galeev AA, Sudan RN (eds) Handbook of plasma physics: volume 1 basic plasma physics, Chapter 3.2. North Holland Publishing Company, pp 451–517
[29] Crighton DG and Leppington FG (1974). Radiation properties of the semi-infinite vortex sheet: the initial-value problem. J Fluid Mech 64(2): 393–414 · Zbl 0298.76045 · doi:10.1017/S0022112074002461
[30] Jones DS and Morgan JD (1972). The instability of a vortex sheet on a subsonic stream under acoustic radiation. Proc Camb Philos Soc 72: 465–488 · Zbl 0251.76036 · doi:10.1017/S0305004100047320
[31] Quinn MC and Howe MS (1984). On the production and absorption of sound by lossless liners in the presence of mean flow. J Sound Vibration 97(1): 1–9 · doi:10.1016/0022-460X(84)90462-0
[32] Rienstra SW (1981). Sound diffraction at a trailing edge. J Fluid Mech 108: 443–460 · Zbl 0473.76061 · doi:10.1017/S0022112081002206
[33] Koch W and Möhring W (1983). Eigensolutions for liners in uniform mean flow ducts. AIAA J 21: 200–213 · Zbl 0526.76081 · doi:10.2514/3.8055
[34] Daniels PG (1985). On the unsteady Kutta condition. Quar J Mech Appl Math 31: 49–75 · Zbl 0389.76018 · doi:10.1093/qjmam/31.1.49
[35] Goldstein ME (1981). The coupling between flow instabilities and incident disturbances at a leading edge. J Fluid Mech 104: 217–246 · Zbl 0463.76053 · doi:10.1017/S0022112081002899
[36] Crighton DG, Innes D (1981) Analytical models for shear-layer feed-back cycles. AIAA81-0061, AIAA Aerospace Sciences Meeting, 19th, St. Louis, MO, 12–15 Jan. 1981
[37] Brandes M, Ronneberger D (1995) Sound amplification in flow ducts lined with a periodic sequence of resonators. AIAA paper 95–126, 1st AIAA/CEAS Aeroacoustics Conference, Munich, Germany, 12–15 June 1995
[38] Aurégan Y, Leroux M, Pagneux V (2005) Abnormal behavior of an acoustical liner with flow. Forum Acusticum 2005, Budapest
[39] Munt RM (1977). The interaction of sound with a subsonic jet issuing from a semi-infinite cylindrical pipe. J Fluid Mech 83(4): 609–640 · doi:10.1017/S0022112077001384
[40] Munt RM (1990). Acoustic radiation properties of a jet pipe with subsonic jet flow: I. The cold jet reflection coefficient. J Sound Vibration 142(3): 413–436
[41] Morgan JD (1974). The interaction of sound with a semi-infinite vortex sheet. Quart J Mech Appl Math 27: 465–487 · Zbl 0319.76058 · doi:10.1093/qjmam/27.4.465
[42] Bechert DW (1980). Sound absorption caused by vorticity shedding, demonstrated with a jet flow. J Sound Vibration 70: 389–405 · doi:10.1016/0022-460X(80)90307-7
[43] Bechert DW (1988). Excitation of instability waves in free shear layers. Part 1. Theory. J Fluid Mech 186(186): 47–62 · Zbl 0643.76039
[44] Howe MS (1979). Attenuation of sound in a low Mach number nozzle flow. J Fluid Mech 91: 209–229 · Zbl 0414.76061 · doi:10.1017/S0022112079000124
[45] Cargill AM (1982). Low-frequency sound radiation and generation due to the interaction of unsteady flow with a jet pipe. J Fluid Mech 121: 59–105 · Zbl 0494.76084 · doi:10.1017/S0022112082001803
[46] Cargill AM (1982). Low frequency acoustic radiation from a jet pipe–a second order theory. J Sound Vibration 83: 339–354 · Zbl 0499.76069 · doi:10.1016/S0022-460X(82)80097-7
[47] Rienstra SW (1983). A small Strouhal number analysis for acoustic wave-jet flow-pipe interaction. J Sound Vibration 86: 539–556 · Zbl 0531.76019 · doi:10.1016/0022-460X(83)91019-2
[48] Rienstra SW (1984). Acoustic radiation from a semi-infinite annular duct in a uniform subsonic mean flow. J Sound Vibration 94(2): 267–288 · doi:10.1016/S0022-460X(84)80036-X
[49] Crighton DG (1985). The Kutta condition in unsteady flow. Ann Rev Fluid Mech 17: 411–445 · doi:10.1146/annurev.fl.17.010185.002211
[50] Peters MCAM, Hirschberg A, Reijnen AJ and Wijnands APJ (1993). Damping and reflection coefficient measurements for an open pipe at low Mach and low Helmholtz numbers. J Fluid Mech 256: 499–534 · doi:10.1017/S0022112093002861
[51] Cummings A (1983). Acoustic nonlinearities and power losses at orifices. AIAA J 22: 786–792 · doi:10.2514/3.8680
[52] Allam S and Åbom M (2006). Investigation of damping and radiation using full plane wave decomposition in ducts. J Sound Vibration 292: 519–534 · doi:10.1016/j.jsv.2005.08.016
[53] Michalke A (1965). On spatially growing disturbances in an inviscid shear layer. J Fluid Mech 23(3): 521–544 · doi:10.1017/S0022112065001520
[54] Jones DS and Morgan JD (1974). A linear model of a finite Helmholtz instability. Proc R Soc Lond A 344: 341–362
[55] Noble B (1958). Methods based on the Wiener–Hopf technique. Pergamon Press, London · Zbl 0082.32101
[56] Heins AE and Feshbach H (1947). The coupling of two acoustical ducts. J Math Phys 26: 143–155 · Zbl 0036.40403
[57] Levine H and Schwinger J (1948). On the radiation of sound from an unflanged circular pipe. Phys Rev (APS) 73(4): 383–406 · Zbl 0039.21302 · doi:10.1103/PhysRev.73.383
[58] Rienstra SW (1986) Hydrodynamic instabilities and surface waves in a flow over an impedance wall. In: Comte-Bellot G, Ffowcs Williams JE (eds) Proceedings IUTAM symposium ’aero- and hydro-acoustics’ 1985 Lyon. Springer-Verlag, Heidelberg, pp 483–490
[59] Abramowitz M and Stegun IA (1964). Handbook of mathematical functions. National Bureau of Standards, Dover Publications Inc., New York · Zbl 0171.38503
[60] Rienstra SW (2006) Impedance models in time domain, including the extended Helmholtz resonator model. AIAA Paper 2006-2686, 12th AIAA/CEAS Aeroacoustics Conference, 8–10 May 2006, Cambridge, MA, USA
[61] Miles JW (1957). On the reflection of sound at an interface of relative motion. J Acoust Soc Am 29(2): 226–228 · doi:10.1121/1.1908836
[62] Kelvin L (1871). Hydrokinetic solutions and observations. Philos Mag 4(42): 362–377
[63] von Helmholtz H (1868). On discontinuous movement of fluids. Philos Mag 4(36): 337–346 · JFM 01.0022.03
[64] Chevaugeon N, Remacle J-F, Gallez X (2006) Discontinuous Galerkin implementation of the extended Helmholtz resonator impedance model in time domain. AIAA paper 2006-2569, 12th AIAA/CEAS Aeroacoustics Conference, Cambridge, MA, 8–10 May 2006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.