The Unified Transform Method (UTM) is applied for the first time to acoustic scattering by poroelastic plates in uniform grazing flow. We extend the UTM framework to incorporate low-Mach-number flow over either a finite poroelastic plate or a composite plate composed of alternating impermeable rigid and poroelastic sections. The method captures edge singularities through tailored basis functions and avoids the kernel factorization difficulties of Wiener–Hopf approaches, while remaining computationally efficient. A flow-dependent Rayleigh conductivity is introduced to quantify how porosity effects are modified by grazing flow under both plane-wave and point quadrupole excitation. Results reveal a Strouhal-controlled regime in which flow can modulate the noise-reducing properties of pores, but beyond this limit the effects of pores are nullified completely. For composite plates, the placement and length of poroelastic inserts and their rigid supports relative to the trailing edge are critical: downstream inserts, when sufficiently long, provide the most effective suppression of trailing-edge noise and elastic insert displacement. These findings demonstrate the UTM as a versatile semi-analytical framework for analyzing noise-control strategies with poroelastic surfaces in flow.
GrahamRR. The silent flight of owls. J Royal Aero Soc1934; 38(286): 837–843.
2.
JaworskiJWPeakeN. Aeroacoustics of silent owl flight. Annu Rev Fluid Mech2020; 52: 395–420.
3.
JaworskiJWPeakeN. Aerodynamic noise from a poroelastic edge with implications for the silent flight of owls. J Fluid Mech2013; 723: 456–479.
4.
AytonLJGillJRPeakeN. The importance of the unsteady Kutta condition when modelling gust–aerofoil interaction. J Sound Vib2016; 378: 28–37.
5.
KisilAAytonLJ. Aerodynamic noise from rigid trailing edges with finite porous extensions. J Fluid Mech2018; 836: 117–144.
6.
HalesADGAytonLJWillsAO, et al.A mathematical model for the interaction of anisotropic turbulence with porous surfaces. J Fluid Mech2024; 1001: A16.
7.
CavalieriAVWolfWRJaworskiJ. Acoustic scattering by finite poroelastic plates. In: 20th AIAA/CEAS Aeroacoustics Conference. Paper AIAA-2014-2459. Aerospace Research Central.
8.
CavalieriAVGWolfWRJaworskiJW. Numerical solution of acoustic scattering by finite perforated elastic plates. Proc Royal Soc A2016; 472(2188): 20150767.
9.
PimentaCWolfWRCavalieriAVG. A fast numerical framework to compute acoustic scattering by poroelastic plates of arbitrary geometry. J Comput Phys2018; 373: 763–783.
10.
GeyerTFSarradjE. Self noise reduction and aerodynamics of airfoils with porous trailing edges. Acoustics2019; 1(2): 393–409.
11.
HerrMRossignolKSDelfsJ, et al.Specification of porous materials for low-noise trailing-edge applications. In:20th AIAA/CEAS Aeroacoustics Conference. Paper AIAA-2014-3041. Aerospace Research Central.
12.
Palleja-CabreSChaitanyaPJosephP, et al.Downstream porosity for the reduction of turbulence-aerofoil interaction noise. J Sound Vib2022; 541: 117–324.
13.
GeyerTFSarradjEFritzscheC. Measuring owl flight noise. In: INTER-NOISE and NOISE-CON Congress and Conference Proceedings, volume 249. Institute of Noise Control Engineering, pp. 183–198.
14.
ClarkIA. Bio-inspired control of roughness and trailing edge noise. PhD Thesis. Virginia Polytechnic Institute and State University, 2017.
15.
ClarkIAAlexanderWNDevenportW, et al.Bioinspired trailing-edge noise control. AIAA J2017; 55(3): 740–754.
16.
MillicanAJClarkIDevenportWJ, et al.Owl-inspired trailing edge noise treatments: acoustic and flow measurements. In: 55th AIAA Aerospace Sciences Meeting. Paper AIAA-2017-1177. Aerospace Research Central.
17.
ClarkIADalyCADevenportW, et al.Bio-inspired canopies for the reduction of roughness noise. J Sound Vib2016; 385: 33–54.
18.
MoreauDJDoolanCJ. Noise-reduction mechanism of a flat-plate serrated trailing edge. AIAA J2013; 51(10): 2513–2522.
19.
OerlemansSFisherMMaederT, et al.Reduction of wind turbine noise using imized airfoils and trailing-edge serrations. AIAA J2009; 47(6): 1470–1481.
20.
NobleB. Methods based on the wiener–Hopf technique. Pergamon Press, 1958.
21.
CrightonDGLeppingtonFG. Scattering of aerodynamic noise by a semi-infinite compliant plate. J Fluid Mech1970; 43(4): 721–736.
22.
RawlinsAD. Acoustic diffraction by an absorbing semi-infinite half plane in a moving fluid. Proc Math Roy Soc Edinb1975; 72: 337–357.
23.
AbrahamsID. Scattering of sound by an elastic plate with flow. J Sound Vib1983; 89: 213–231.
24.
KisilAVAbrahamsIDMishurisG, et al.The wiener–hopf technique, its generalizations and applications: constructive and approximate methods. Proc Royal Soc A2021; 477(2254): 20210533.
25.
DanieleVG. On the solution of two coupled Wiener–Hopf equations. SIAM J Appl Math1984; 44(4): 667–680.
26.
AytonLJ. Acoustic scattering by a finite rigid plate with a poroelastic extension. J Fluid Mech2016; 791: 414–438.
27.
GeyerTFSarradjEGieslerJ. Application of a beamforming technique to the measurement of airfoil leading edge noise. Adv Acou Vib2012; 2012: 1–16.
28.
MoreauDBrooksLDoolanC. On the noise reduction mechanism of a flat plate serrated trailing edge at low-to-moderate Reynolds number. In: 18th AIAA/CEAS Aeroacoustics Conference. Paper AIAA-2012-2186. Aerospace Research Central.
29.
SandbergRJonesL. Direct numerical simulations of airfoil self-noise. Procedia Eng2010; 6: 274–282.
30.
TerunaCAvalloneFCasalinoD, et al.Numerical investigation of leading edge noise reduction on a rod-airfoil configuration using porous materials and serrations. J Sound Vib2021; 494: 115880.
31.
FokasASPelloniB. Unified transform for boundary value problems. Society for Industrial and Applied Mathematics, 2014.
32.
FokasASKapaevAA. On a transform method for the Laplace equation in a polygon. IMA J Appl Math2003; 68(4): 355–408.
33.
ColbrookMJAytonLJFokasAS. The unified transform for mixed boundary condition problems in unbounded domains. Proc Royal Soc A2019; 475: 20180605.
34.
AytonLJColbrookMJFokasAS. The unified transform: a spectral collocation method for acoustic scattering. In: 25th AIAA/CEAS Aeroacoustics Conference. Paper AIAA-2019-2528. Aerospace Research Central.
35.
ColbrookMAytonL. A spectral collocation method for acoustic scattering by multiple elastic plates. J Sound Vib2019; 461: 114904.
36.
NaqviSHalesADGAytonLJ, et al.An efficient and versatile semi-analytical method for acoustic wave scattering by periodic structures in flow. In: AIAA Aviation Forum and Ascend 2025. Paper AIAA-2025-3363. Aerospace Research Central.
NaqviSBAytonLJ. A periodic extension to the Fokas method for acoustic scattering by an infinite grating. Acoustics2025; 7(1): 5.
39.
NaqviSB. Acoustic scattering by gratings. PhD Thesis. University of Cambridge, 2025.
40.
ColbrookMJPriddinMJ. Fast and spectrally accurate numerical methods for perforated screens (with applications to Robin boundary conditions). IMA J Appl Math2020; 85: 790–821.
41.
ScottMI. The Rayleigh conductivity of a circular aperture in the presence of a grazing flow. Master’s thesis. Boston University, 1995.
42.
HoweMS. Acoustics of fluid-structure interactions. Cambridge monographs on mechanics. Cambridge University Press, 1998.
43.
HoweMS. The damping of flexural and acoustic waves by a bias-flow perforated elastic plate. Eur J Appl Math1995; 6: 307–328.
44.
GraceSHoranKPHoweMS. The influence of shape on the Rayleigh conductivity of a wall aperture in the presence of grazing flow. J Fluid Struct1998; 12: 335–351.
45.
DugundjiJDowellEPerkinB. Subsonic flutter of panels on continuous elastic foundations. AIAA J1963; 1(5): 1146–1154.
46.
KorneckiADowellEO’BrienJ. On the aeroelastic instability of two-dimensional panels in uniform incompressible flow. J Sound Vib1976; 47: 163–178.
47.
HajianRJaworskiJW. Noncirculatory fluid forces on panels with porosity gradients. AIAA J2019; 57(2): 870–875.
48.
HajianRJaworskiJW. Porous panels with fixed ends lose stability by divergence. J Fluid Struct2020; 93: 102823.
49.
HajianRJaworskiJGraceSM. Acoustic emission from one-dimensional vibrating porous panels in a single-sided flow. In: 2018 AIAA/CEAS Aeroacoustics Conference. Paper AIAA-2018-2962. Aerospace Research Central.
50.
RayleighL. The theory of sound, volume two. In: Cambridge Library Collection - Physical Sciences. Cambridge University Press, 1877.
51.
HoweMSScottMISipcicSR. The influence of tangential mean flow on the Rayleigh conductivity of an aperture. Proc Royal Soc London Series A1996; 452(1953): 2303–2317.
52.
LiuQLiuYJiangH, et al.Acoustic scattering by a finite plate with a poroelastic extension using the unified transform method. J Sound Vib2022; 522: 116677.
53.
HalesADGAytonLJJiangC, et al.A mathematical model for the interaction of anisotropic turbulence with a rigid leading edge. J Fluid Mech2023; 970: A29.
54.
BartonPGRawlinsAD. Diffraction by a half-plane in a moving fluid. Q J Mech Appl Math2005; 58(3): 459–479.
55.
SpenceEA.Boundary Value Problems for Linear Elliptic PDEs. PhD Thesis, University of 705Cambridge, 2010.
56.
CrightonDG. The Kutta condition in unsteady flow. Annu Rev Fluid Mech1985; 17: 411–445.
57.
ColbrookMJFlyerNFornbergB. On the Fokas method for the solution of elliptic problems in both convex and non-convex polygonal domains. J Comput Phys2018; 374: 996–1016.
58.
ColbrookMJ. Extending the unified transform: curvilinear polygons and variable coefficient pdes. IMA J Numer Anal2020; 40: 976–1004.
59.
NaqviSBAytonLJ. Homogenisation of perforated plates. In: 28th AIAA/CEAS Aeroacoustics Conference. Paper AIAA-2022-2902. Aerospace Research Central.
60.
PriddinMJParuchuriCCJosephP, et al.A semi-analytic and experimental study of porous leading edges. In: 25th AIAA/CEAS Aeroacoustics Conference. Paper AIAA-2019-2552.
61.
HalesADG. An analytical investigation into novel poroelastic boundary conditions for noise-reducing metamaterials. In: AIAA Aviation Forum and Ascend 2025. Paper AIAA-2025-3363. Aerospace Research Central.
