1. Introduction
Trailing-edge (TE) noise represents a substantial component of airfoil self-noise, particularly in applications such as wind turbines, turbomachinery blades and airfoils. The broadband noise is generated when a turbulent boundary layer passes over the sharp TE, converting a small fraction of the turbulent energy into acoustic energy. Recent studies have demonstrated some effective bio-inspired noise control techniques (Graham Reference Graham1934; Jaworski & Peake Reference Jaworski and Peake2020), including serrations (Howe Reference Howe1991b ; Moreau & Doolan Reference Moreau and Doolan2013; Lyu, Azarpeyvand & Sinayoko Reference Lyu, Azarpeyvand and Sinayoko2016; Huang Reference Huang2017; Ayton Reference Ayton2018), finlets (Clark et al. Reference Clark, Alexander, Devenport, Glegg, Jaworski, Daly and Peake2017; Bodling & Sharma Reference Bodling and Sharma2019; Pan & Ye Reference Özyörük, Long and Jones2025), rough elements (Yuan et al. Reference Yeh and Taira2025) and perforations/porous treatments (Howe Reference Howe1979; Shen et al. Reference Schulze and Sesterhenn2022). Among these, tailored porous media (Schulze & Sesterhenn Reference Schmidt and Towne2013; Zhou et al. Reference Zhao, Liu and Lu2018; Zhang & Chong Reference Yuan, Alva, de Araújo, Cavalieri and Hanifi2020; Shen et al. Reference Schulze and Sesterhenn2022; Sun et al. Reference Suman, Sundaram, Puttam, Sengupta and Sengupta2022) show great potential for suppressing broadband noise.
1.1. Noise reduction with porous trailing edges
The early theoretical work was instrumental in shaping our understanding and predictive capabilities of the TE noise with porous TEs. Howe (Reference Howe1979) derived the scattering acoustic pressure from a perforated TE based on the inviscid wave equation. By comparing the acoustic propagation characteristics under different surface impedances, it was found that a gradually varying impedance experienced by a turbulent eddy is more favourable than the impedance jump associated with a uniform distribution; however, further optimisation of the functional form of the impedance remains imperative. Jaworski & Peake (Reference Jaworski and Peake2013) explored the acoustic interaction of a turbulent eddy with a semi-infinite poroelastic edge and identified its excellent performance for noise reduction. They analysed the scaling dependence on the characteristic Mach number
$\textit{Ma}$
, where the acoustic power scales with
$\textit{Ma}^6$
for a porous edge and
$\textit{Ma}^7$
for an elastic edge. Both are less efficient than a rigid TE, which has the well-known
$\textit{Ma}^5$
dependence. The different scaling laws further suggest that the combination of porosity and flexibility may be the key factors to the silent flight of owls and other creatures. However, the semi-infinite plate model fails to account for the acoustic interaction between the leading and trailing edges, potentially leading to discrepancies between the predicted and actual airfoil noise. To address this, Ayton (Reference Ayton2016) investigated the finite chord length effects of poroelastic edge based on the Padé approximations to Wiener–Hopf factorisations. The results demonstrate that the extent of the poroelastic part can significantly affect noise radiation, where a longer extension is more beneficial for inducing more potential resonant modes. Another efficient Wiener–Hopf technique in solving the TE scattering problem by finite porous extension was performed by Kisil & Ayton (Reference Kisil and Ayton2017), where the interaction of impermeable–permeable junction and the TE was found to play an important role in reducing low-frequency noise in the far field, characterised by compact radiation behaviour. However, the interference of the junction has a negative effect in the high-frequency range.
The theoretical studies often rely on substantial simplifications, whereas high-fidelity numerical simulations and experimental techniques offer more reliable noise predictions in complex scenarios, particularly those involving viscous effects, non-frozen turbulence, finite airfoil thickness and boundary-layer shear. Ali, Azarpeyvand & Da Silva (Reference Ali, Azarpeyvand and Da Silva2018) investigated the effects of porous blunt TEs on flow control and vortex shedding modulation experimentally. The results demonstrated that the flow–porous scrubbing induces flow penetration into the porous region and discharges into the near wake. The vortex shedding is delayed and weakened, which is closely related to the internal quasi-periodic recirculating flow. The energetic frequency content indicates that the reduction in spanwise coherence results in suppressed tonal noise at the vortex shedding frequency, accompanied by a broadband hump at higher frequencies due to flow modifications within the porous media. A subsequent anechoic wind tunnel experiment by Zhou, Zhong & Zhang (Reference Zhou, Qiu, Li and Liu2021) focused on the noise from velvety structures on the TE. It was observed that the vortex shedding at the TE was significantly suppressed, and the TE noise spectrum was markedly altered, characterised by an increase in noise just below the critical frequency (the Strouhal number
$St$
typically exceeds 10 when scaled by the airfoil chord length and free-stream velocity) and a reduction in high-frequency noise. Furthermore, by estimating the source terms of the Blake model (Blake Reference Blake1986; Parchen Reference Pan and Ye1998), the important contributions of the wall-normal velocity gradient and turbulent fluctuations near the wall surface to noise reduction were evaluated quantitatively. To enable a more comprehensive evaluation of the noise reduction characteristics of porous TEs on airfoils, Teruna et al. (Reference Teruna, Avallone, Ragni, Rubio-Carpio and Casalino2020) performed a very large eddy simulation to explore the noise emission of an open-cell metal-foam TE. The results demonstrate that fully porous media, as opposed to the blocked-porous design, yield a noise attenuation of 9 dB within the low-frequency domain (
$4\leqslant St \leqslant 8$
). Beamforming analysis revealed that the dominant acoustic source was located at the TE for the solid and blocked-porous TEs, whereas it shifted to the solid–porous junction for the fully porous TEs. Further analysis indicates that, while the spectral characteristics of the wall-pressure fluctuations are relatively similar, the fully porous TE leads to a gradual decrease in wall impedance along the flow direction, which, in turn, results in a lower scattering efficiency. In a subsequent study, Teruna et al. (Reference Teruna, Avallone, Ragni and Casalino2021b
) eschewed a modelling approach in favour of directly resolving the flow within 3D-printed porous media. The results demonstrate that, although porous media can achieve effective low-frequency noise reduction (
$4\leqslant St \leqslant 8$
), a slight increase in the high-frequency range arises due to impedance discontinuity at the solid–porous junction. Notably, the installation of a solid plate at 84 %–96 % of the airfoil chord length attenuated the low-frequency noise reduction while leaving the mid- (
$8\leqslant St \leqslant 16$
) and high-frequency (
$16\leqslant St \leqslant 32$
) components largely unaffected. Due to the combined effects of surface roughness and permeability, both the velocity and pressure fluctuations at the wall were substantially enhanced. When the local porous medium thickness was less than twice the entrance length (a normal distance from the porous medium’s interior to its surface, where the turbulence intensity is comparable to that of the external flow) (Baril et al. Reference Baril, Mostafid, Lefebvre and Medraj2008), the pressure release process at the TE resulted in reduced acoustic radiation efficiency. Additionally, the acoustic radiation was mainly affected by the partial interference between distributed sources across multiple locations.
Building upon the fundamental understanding of the noise reduction mechanism in porous TEs, the optimal design of porous media has attracted considerable research interest in recent years. Early experimental studies by Hayden, Kadman & Chanaud (Reference Hayden, Kadman and Chanaud1972) and Hayden (Reference Hayden1976a ,Reference Hayden b ) demonstrated the efficacy of variable impedance TEs in mitigating the noise of upper surface blown configurations, which was further supported through the theoretical analysis of Howe (Reference Howe1979). Subsequently, a discrete-adjoint optimisation scheme was adopted by Schulze & Sesterhenn (Reference Schmidt and Towne2013) to reduce two-dimensional blunt plate TE noise using an optimised porous medium. After five iterations, a maximum noise reduction of up to 19 dB was achieved, and the optimal permeability coefficient spanned multiple scales while exhibiting significant anisotropy. Zhou et al. (Reference Zhao, Liu and Lu2018) split the porous medium region at the TE into distinct spanwise strips and applied the same optimisation strategy to obtain the best porous parameters. The optimisation results varied when starting with different baseline cases, indicating the multi-modal nature of the parameter space. Nevertheless, the optimal results still demonstrated that a streamwise-increasing porosity and viscous permeability, and a decreasing thermal permeability, could achieve a noise reduction of up to 12 dB. Recently, the advantages of a non-uniform distribution of resistance have been further elucidated with numerical (Liu, Chen & Hu Reference Liu, Chen and Hu2019; Xiao et al. Reference Xia and Mohseni2025) and experimental methods (Wang et al. Reference Vinuesa, Negi, Atzori, Hanifi, Henningson and Schlatter2021), reaffirming its substantial potential for noise reduction.
1.2. Kutta condition in trailing-edge noise
The Kutta–Joukowsky hypothesis, commonly known as the Kutta condition (Giesing Reference Giesing1969; Poling & Telionis Reference Parchen1986; Xia & Mohseni Reference Wolf, Lutz, Würz, Krämer, Stalnov and Seifert2017), asserts that the flow from the upper and lower sides must leave the TE smoothly, which prevents singularities in velocity and pressure near the sharp edge. For unsteady turbulent flows, the Kutta condition is enforced. The pressure or velocity jump across the TE affects vortex shedding and the conversion of a portion of turbulent energy into acoustic energy, resulting in the intense radiation of noise (Crighton Reference Crighton1985). Previous studies (Amiet Reference Amiet1990) have indicated that the flow at the TE of an infinitely thin and symmetric plate inherently satisfies the Kutta condition. For the complex configurations, Ayton, Gill & Peake (Reference Ayton, Gill and Peake2016) compared the analytical and numerical results of the noise radiated by an airfoil. The results illustrated that neglecting the unsteady Kutta condition induces unphysical scattering of a pressure singularity, which mainly affects the noise radiation upstream, especially when the airfoil configuration has finite thickness. Moreover, the pressure release process (Herr et al. Reference Herr, Rossignol, Delfs, Lippitz and Mößner2014; Delfs et al. Reference Delfs, Faßmann, Lippitz, Lummer, Mößner, Müller, Rurkowska and Uphoff2014; Teruna et al. Reference Teruna, Avallone, Ragni and Casalino2021b ), reported in previous studies on porous TEs, is intrinsically linked to the Kutta condition. Specifically, the presence of a porous medium induces flow interaction between the suction and pressure surfaces in the vicinity of the TE, thereby attenuating the pressure jump across the upper and lower surfaces and resulting in a lower acoustic scattering efficiency.
1.3. Outline
Despite variations in the noise reduction levels of porous TEs under different flow configurations, two main design principles can be summarised: (i) the flow impedance should vary gradually towards higher viscous permeability (Howe Reference Howe1979; Schulze & Sesterhenn Reference Schmidt and Towne2013; Wang et al. Reference Vinuesa, Negi, Atzori, Hanifi, Henningson and Schlatter2021), (ii) the pressure jump between the upper and lower surfaces should be minimised (Teruna et al. Reference Teruna, Avallone, Ragni, Rubio-Carpio and Casalino2020, Reference Teruna, Avallone, Ragni and Casalino2021b ). Our study aims to leverage these principles by designing an effective impedance TE configuration that minimises the pressure jump, consistent with the unsteady Kutta condition. Given the inherent complexity of flow–porous interaction, high-fidelity numerical simulations with relatively higher spatial-temporal resolution of the near-wall turbulence field are indispensable. This critical resolution enables us to systematically investigate the wall-pressure jump and the TE vortex shedding process, thus providing a comprehensive acoustic source characterisation consistent with the Kutta condition. By dissecting the complex interplay of multiple noise reduction mechanisms (including flow modification, acoustic interference and source suppression) and quantitatively isolating the most dominant factors, this comprehensive investigation seeks to fully elucidate the physical mechanism and yield robust, fine-grained insights for optimal low-noise design strategies.
The remainder of this paper is structured as follows. The governing equations, acoustic analogy, a newly improved wall-pressure spectrum model, vortex-sound diagnostic methods and numerical cases are briefly introduced in
$\S$
2. Section 3.1 compares the acoustic performance for different impedance TEs. In
$\S$
3.2, we discuss the changes in flow structures and turbulence statistics, while
$\S$
3.3 presents an analysis of the dynamic evolution of the coherent structures. Next, the wavenumber–frequency spectrum of the wall-pressure jump and the related edge scattering are compared in
$\S$
3.4 to elucidate the potential noise reduction mechanism. Section 3.5 systematically compares the vortex-sound sources across the turbulent boundary layer to obtain deeper insights. Finally, the main conclusions are summarised.
2. Numerical procedures
2.1. The governing equations
The fully compressible Navier–Stokes equations are solved using implicit large eddy simulation for an NACA-0012 airfoil with solid and impedance TEs. The equations are presented in the non-dimensional conservation form as
\begin{align} \frac {\partial \rho }{\partial t} +\frac {\partial }{\partial x_j}\left (\rho u_j\right ) &= 0, \end{align}
\begin{align} \frac {\partial }{\partial t} (\rho u_i)+\frac {\partial }{\partial x_j}(\rho u_i u_j) &= -\frac {\partial p}{\partial x_i}+\frac {\partial }{\partial x_j}(\tau _{\textit{ij}}), \end{align}
\begin{align} \frac {\partial }{\partial t}(\rho E)+\frac {\partial }{\partial x_j}(\rho u_j E) &= -\frac {\partial }{\partial x_j} (pu_j)+\frac {\partial }{\partial x_j}(\tau _{\textit{ij}}u_i)- \frac {\partial q_j}{\partial x_j}, \end{align}
where
$\rho$
is the density,
$u_i$
is the velocity component,
$p$
is the pressure and the total energy is defined as
$E = p/[\rho (\gamma -1)]+u_ju_j/2$
. The variables are non-dimensionalised by the far-field density
$\rho _\infty$
, the speed of sound
$c_\infty$
and the airfoil chord length
$L$
. The viscous stress
$\tau _{\textit{ij}}$
and heat flux vector
$q_j$
are given by
\begin{equation} \tau _{\textit{ij}} = \frac {\mu \textit{Ma}}{\textit{Re}}\left(\frac {\partial u_i}{\partial x_j}+\frac {\partial u_j}{\partial x_i}-\frac {2}{3}\frac {\partial u_m}{\partial x_m}\delta _{\textit{ij}}\right)\!, \quad \quad q_j = -\frac {\mu \textit{Ma}}{(\gamma -1)\textit{PrRe}}\frac {\partial T}{\partial x_j}, \end{equation}
where
$\delta_{ij}$
is the Kronecker delta,
$\gamma =1.4$
is the specific heat ratio,
$\textit{Pr}=0.72$
is the Prandtl number,
$\textit{Ma}=U_\infty /c_\infty$
is the free-stream Mach number,
$\textit{Re}=\rho _\infty U_\infty L/\mu _\infty$
is the Reynolds number and
$T$
denotes temperature. The dynamic viscosity,
$\mu$
, is computed using the Sutherland law, and the ideal gas equation of state
$p=\rho T/\gamma$
is employed.
The adiabatic time-domain impedance boundary condition (TDIBC) (Tam & Auriault Reference Sun, Wang, Du and Sun1996; Özyörük et al. Reference Zhou, Zhong and Zhang1998; Fung & Ju Reference Fung and Ju2004; Richter et al. Reference Ribeiro, Casalino, Fares and Choudhari2011; Scalo, Bodart & Lele Reference Sano, Abreu, Cavalieri and Wolf2015; Lin, Scalo & Hesselink Reference Lin, Scalo and Hesselink2016; Chen & Scalo Reference Chen and Scalo2021b ) is applied in the wall-normal direction, and the no-slip boundary condition is used in the wall-tangential direction to the controlled TEs, given by
\begin{equation} \left \{ \begin{aligned} \hat {p}&=Z(\omega )\hat {v}_n, Z(\omega ) = R+i\frac {1+R}{2\zeta }(\omega /\omega _r-\omega _r/\omega ),\\ u_t & = w = 0, \partial T/\partial n=0, \end{aligned} \right . \end{equation}
where
$Z(\omega)$
is the impedance function,
$R$
is the resistance,
$\omega _r=2\pi f_r$
is the characteristic angular frequency (
$f_r$
is the resonance frequency) and
$\zeta$
is the damping ratio for the porous material (implementation details are provided in Appendix B). Here,
$u_t$
denotes the tangential velocity component,
$v_n$
is the normal component and the symbol
$\hat {\square }$
represents the Fourier transform of a time-varying quantity.
The flow over a NACA-0012 airfoil is simulated numerically using the in-house high-order finite-difference solver Aero3DX (Hu et al. Reference Hu, Wan, Ye, Sun and Lu2025a , Reference Hu, Wan, Zhang, Sun and Lub ). The convection terms are discretised using a low-dissipation fifth-order alternative upwind scheme, while the viscous terms are computed with a sixth-order central difference scheme. Interface fluxes are evaluated by means of a low-dissipation Harten-Lax-van Leer-Contact approximate Riemann solver. The third-order total variation diminishing Runge–Kutta scheme (Gottlieb & Shu Reference Gottlieb and Shu1998) is adopted for time integration. These enhancements allow the present numerical scheme to accurately resolve small-scale turbulent structures and propagating waves. The present computations are performed on modern high-performance graphics processing unit clusters.
2.2. Ffowcs Williams–Hawkings acoustic analogy
The Ffowcs Williams–Hawkings acoustic analogy method (Ffowcs Williams & Hawkings Reference Ffowcs Williams and Hawkings1969; Brentner & Farassat Reference Brentner and Farassat1998) is adopted to obtain the far-field noise radiated by solid or impedance TEs. The convective wave equation for pressure fluctuation is expressed as
\begin{align} \left [ \frac {1}{c_\infty ^2}{D^2}-\boldsymbol{\nabla} ^2 \right ] \left [ H(f)\rho ^{\prime } \right ] &=\left(\frac {\partial }{\partial t}+U_{\infty j}\frac {\partial }{\partial x_j}\right)\big[Q_k n_k\delta (f)\big] -\frac {\partial }{\partial x_i}\big[L_{\textit{ij}}n_j \delta (f)\big]\nonumber\\&\quad +\frac {\partial }{\partial x_i \partial x_j}\big[H(f)\mathcal{T}_{\textit{ij}}\big], \end{align}
and
\begin{equation} \begin{aligned} Q_j &= \big[\rho \big(u_{\!j}^{\prime } + U_{\infty j}-v_j\big)+\rho _\infty \big(v_j - U_{\infty j}\big)\big],\\ L_{\textit{ij}}&=\big[\rho u_i^{\prime }\big(u_{\!j}^{\prime }+U_{\infty j}-v_j\big) +\big(p-p_\infty \big)\delta _{\textit{ij}}-\tau _{\textit{ij}}\big],\\ \mathcal{T}_{\textit{ij}}&=\rho u_i^{\prime } u_{\!j}^{\prime }+\big[(p-p_\infty )-c_\infty ^2(\rho -\rho _\infty )\big]\delta _{\textit{ij}}-\tau _{\textit{ij}},\\ \end{aligned} \end{equation}
where
$Q_j$
is the monopole source (or thickness source),
$L_{\textit{ij}}$
is the dipole source (or loading source) and
$\mathcal{T}_{\textit{ij}}$
is the quadrupole source. In this equation,
$H$
is the Heaviside function,
$n_i$
is the component of the unit normal vector on the permeable surface,
$U_{\infty j}$
denotes the free-stream velocity component in the far field,
$v_j$
is the surface velocity and
$\tau _{\textit{ij}}$
is the viscous stress. The superscript
$\square ^\prime$
denotes the fluctuating part of a variable.
For turbulent flow at a low Mach number, the quadrupole noise is typically not computed directly owing to the lower scattering efficiency (Howe Reference Howe2003), and the contribution from the viscous stress is also neglected. With the free space Green’s function, the final solution in the frequency domain is expressed as (Ghorbaniasl et al. Reference Ghorbaniasl, Huang, Siozos-Rousoulis and Lacor2015)
\begin{align} 4\pi p^{\prime }_T(\boldsymbol{y}, \omega ) =&\int _{S}\frac {\left(1-U_{\infty j}\widetilde R_j/c_\infty \right)}{R^*} i\omega \widetilde {Q}_k(\boldsymbol{x}, \omega ) n_k e^{-ikR} \text{d}\boldsymbol{S}\nonumber \end{align}
\begin{align} &- U_{\infty j} \int _{S}\frac {\widetilde R_j}{{R^*}^2}\widetilde {Q}_k(\boldsymbol{x}, \omega ) n_k e^{-ikR}\text{d}\boldsymbol{S}, \end{align}
\begin{align} 4\pi p^{\prime }_L(\boldsymbol{y}, \omega ) =&\int _{S}\left (\frac {\widetilde R_i}{R^*}\right ) ik\widetilde {L}_{\textit{ij}}(\boldsymbol{x}, \omega ) n_j e^{-ikR} \text{d}\boldsymbol{S}+\int _{S}\left (\frac {\widetilde R^*_i}{{R^*}^2}\right ) \widetilde {L}_{\textit{ij}}(\boldsymbol{x}, \omega ) n_j e^{-ikR} \text{d}\boldsymbol{S}, \end{align}
where
\begin{equation} \begin{aligned} R &= \frac {-U_{\infty j}(x_1-y_1)/c_\infty +R^*}{\chi ^2},\\ R^* &= \sqrt {(x_1-y_1)^2+\chi ^2(x_2-y_2)^2+\chi ^2(x_3-y_3)^2},\\ \widetilde {R}_i &= \partial R/\partial x_i, \widetilde {R}^*_i = \partial R^*/\partial x_i,\\ \chi ^2 &= 1-\textit{Ma}^2,\\ \end{aligned} \end{equation}
where
$p_T^\prime$
is the thickness noise contributed by the monopole source and
$p^\prime _L$
is the loading noise contributed by the dipole source. Here,
$\boldsymbol{x}$
and
$\boldsymbol{y}$
are the locations of the source on the permeable surface and the observer in the far field, respectively.
2.3. A newly improved wavenumber–frequency spectrum of wall-pressure jump
For wall turbulence, the energy of surface pressure fluctuations typically satisfies specific wavenumber–frequency relationships, such as convective ridges and acoustic ridges (Gloerfelt & Berland Reference Gloerfelt and Berland2013). When the wall surface is modelled by an impedance boundary, additional surface waves (Rienstra Reference Richter, Hay, Schönwald, Busse and Thiele1986) will be induced. Traditional spectral models are predominantly applied to the infinite flat plate. To facilitate a newly improved spectrum predictive model for the spectral characteristics near the airfoil impedance TE, the following key assumptions are necessary: (i) the rear section of the airfoil may be approximated as an infinite plate, provided the frequency is sufficiently high that the acoustic wavelength remains less than the streamwise length of the impedance wall (Howe Reference Howe1991a
); (ii) the convection velocity across the boundary layer is assumed to be uniformly distributed over a limited streamwise extent. For waves propagating in a two-dimensional
$x{-}y$
plane (
$x$
and
$y$
can be considered the wall-tangential and wall-normal coordinates, respectively), the fluctuation field
$ [p^{\prime }, u^{\prime }, v^{\prime } ]$
is governed by the convective Helmholtz equation (Rienstra & Hirschberg Reference Rienstra2004)
\begin{align} \left (\frac {\partial }{\partial t} + \textit{Ma}\frac {\partial }{\partial x}\right )p^{\prime } + \boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{u}^\prime &= 0, \end{align}
\begin{align} \left (\frac {\partial }{\partial t} + \textit{Ma}\frac {\partial }{\partial x}\right )\boldsymbol{u}^\prime + \boldsymbol{\nabla }p^{\prime } &= 0. \end{align}
By combining (2.10a
) and (2.10b
) and eliminating
$\boldsymbol{u}^\prime$
, the governing equation for the pressure fluctuation
$p^\prime$
is derived as
\begin{equation} \left (\frac {\partial }{\partial t} + \textit{Ma}\frac {\partial }{\partial x}\right )^2 p^{\prime } - \boldsymbol{\nabla} ^2 p^{\prime } = 0. \end{equation}
Assuming a harmonic wave
$p^\prime =\hat {p}(x,y)e^{i\omega t}$
for the fluctuation field, one obtains
\begin{equation} \left (ik + \textit{Ma}\frac {\partial }{\partial x}\right )^2 \hat {p} - \boldsymbol{\nabla} ^2 \hat {p} = 0, \end{equation}
where
$k=\omega /c_\infty$
denotes the wavenumber. By further assuming normal modes for
$\hat {p}(x,y)$
, one can derive the following expressions:
\begin{equation} \hat {p} = A e ^{-ik\alpha x-ik\varGamma y}, \,\,\, \hat {v} = \frac {A \varGamma }{\rho _\infty (1-\alpha \textit{Ma})} e ^{-ik\alpha x-ik\varGamma y}. \end{equation}
Substituting (2.13) into (2.12) yields
\begin{equation} \big(1-\textit{Ma}^2\big)\alpha ^2 + 2\alpha \textit{Ma} + (\varGamma ^2-1)=0, \end{equation}
or equivalently,
\begin{equation} \frac {\big(\alpha +\textit{Ma}/\chi ^2\big)^2}{\big(1/\chi ^2\big)^2} + \frac {\varGamma ^2}{(1/\chi )^2}=0, \end{equation}
where
$\chi =\sqrt {1-\textit{Ma}^2}$
is the Prandtl–Glauert parameter (Gloerfelt & Berland Reference Gloerfelt and Berland2013).
The streamwise modes can then be identified as
\begin{equation} \alpha = \frac {-\textit{Ma} \pm \sqrt {1-\big(1-\textit{Ma}^2\big)\varGamma ^2}}{1-\textit{Ma}^2}. \end{equation}
If
$\varGamma =0$
, then
\begin{equation} \alpha = (\textit{Ma} \pm 1)^{-1}, \end{equation}
which represents the downstream supersonic and upstream subsonic wave.
A surface wave denotes a class of waves that decay exponentially away from the impedance wall surface (Rienstra Reference Richter, Hay, Schönwald, Busse and Thiele1986; Rienstra & Hirschberg Reference Rienstra2004; Hu et al. Reference Hu, Wan, Ye, Sun and Lu2025a ). From the impedance condition (2.5) and (2.13), the following relation is obtained:
\begin{equation} \rho (1-\alpha \textit{Ma}) - Z\varGamma \xi =0, \,\,\text{or}\,\,\varGamma = {\rho (1-\alpha \textit{Ma})}/({Z\xi }), \end{equation}
where
$\xi =-1$
for the suction side and
$\xi =1$
for the pressure side. Substituting (2.18) into (2.16) yields
\begin{equation} \alpha = \frac {-\textit{Ma}(Z^2-1) \pm \sqrt {Z^2(Z^2-1)}}{(1-\textit{Ma}^2)Z^2 + \textit{Ma}^2}. \end{equation}
Consequently,
\begin{equation} \omega \rightarrow 0 \,\text{and} \,R \rightarrow 0 \Longrightarrow \alpha \rightarrow 1/(\textit{Ma}\pm 1), \end{equation}
where the acoustic ridges are recovered.
Moreover, when the frequency approaches the resonance frequency and the resistance tends to vanish, a pair of repeated roots emerges, i.e.
\begin{equation} \omega \rightarrow \omega _r \,\text{and}\, \,R \rightarrow 0 \Longrightarrow \alpha _1 \rightarrow \alpha _2. \end{equation}
Finally, the streamwise wavenumber
$k_x$
can be expressed as
\begin{equation} k_x = k\alpha = \frac {\omega }{c_\infty } \boldsymbol{\cdot }\frac {-\textit{Ma}(Z^2-1) \pm \sqrt {Z^2(Z^2-1)}}{(1-\textit{Ma}^2)Z^2 + \textit{Ma}^2}. \end{equation}
By requiring the solution to exhibit both decay in the normal direction and streamwise instability, the specific solution corresponding to the surface wave can be isolated from the two solutions of the above expression. Relevant details are provided in our previous work (Hu et al. Reference Hu, Wan, Ye, Sun and Lu2025a ).
It is worth noting that although the above derivation is based on pressure fluctuations on a single side of the wall, the pressure jump on the wing surface, defined by
\begin{equation} \overset {\frown } p(x, z, t) = p_{ss}(x, z, t)-p_{ps}(x, z, t), \end{equation}
follows the same pattern due to the statistical symmetry of the flow, where
$p_{ss}$
is the wall pressure on the suction side and
$p_{ps}$
is the wall pressure on the pressure side.
To obtain the wavenumber–frequency spectrum based on large eddy simulation data, a spatio-temporal Fourier transform is performed on the wall-pressure jump, yielding
\begin{equation} \begin{aligned} & \hat {p}\left (k_{x}, k_{z}, \omega \right ) \quad =\iiint \overset {\frown } p\left (x, y, t\right ) \textit{e}^{-\textit{i} k_{x} x-\textit{i} k_{z} z+\textit{i} \omega t}\,\text{d}x \,\text{d}z \,\text{d}t, \end{aligned} \end{equation}
and the wavenumber–frequency spectrum is then defined by
\begin{equation} \varPhi _{\overset {\frown } p\overset {\frown } p}\left (k_{x}, k_{z}, \omega \right )=\mathcal{E}\left \{\lim _{L_{x}, L_{z}, L_t \rightarrow \infty } \frac {\hat {p} \hat {p}^*}{L_{x} L_{z} L_t}\right \}, \end{equation}
where
$\mathcal{E}$
is the ensemble average operator,
$\hat {p}^*$
is the complex conjugate of
$\hat {p}$
,
$L_t$
is the total time,
$L_x$
is the wall-tangential extent for spectrum analysis and
$L_z$
represents the spanwise width.
2.4. Noise generation with near-field vortex dynamics
Although the radiation efficiency of vortex-related sound sources is lower than that of wall dipole sources, it is crucial to incorporate the contribution of the vortex-associated component for a comprehensive and accurate assessment of noise reduction. Nevertheless, the definition of vortex-related sound sources is not unique (Lighthill Reference Lighthill1952; Powell Reference Poling and Telionis1964; Möhring Reference Möhring1978; Howe Reference Howe1998, Reference Howe2003; Mao et al. Reference Mao, Kang, Wu, Yu, Gao, Su and Lu2020; Arya & De Reference Arya and De2021; Zhao et al. Reference Zhang and Chong2025). In this study, we primarily consider three distinct definitions of the source terms, with the expectation of attaining consistent insights to some extent.
The first source term
$S_1 = \omega _i \omega _j \partial u_i/\partial x_j$
, or the vortex stretching and tilting source, can be derived from the transport equation of enstrophy (Zhao, Liu & Lu Reference Zhao, Ding, Weng, Zhou, Dong and Yang2020; Suman et al. Reference Spiropoulos, Margnat, Valeau and Jordan2022) as
\begin{equation} \begin{aligned} D\left(\frac {1}{2}\omega _i \omega _i\right) = & \underbrace {\omega _i\omega _j \frac {\partial u_i}{\partial x_j}}_{S_1\text{:vortex stretching and tilting}} \underbrace {- \omega _i \omega _i \frac {\partial u_k}{\partial x_k}}_{\text{compressibility effect}}\\ &+ \underbrace {\frac {1}{\rho ^2}\epsilon _{\textit{ijk}} \omega _i \frac {\partial \rho }{\partial x_j}\frac {\partial p}{\partial x_k}}_{\text{baroclinic term}} + \underbrace {\frac {1}{\textit{Re}}\epsilon _{\textit{ijk}}\omega _i \frac {\partial }{\partial x_j}{\frac {1}{\rho }\frac {\partial \tau _{lk}}{\partial x_l}}}_{\text{viscous term}}, \end{aligned} \end{equation}
where
$D={\partial }/{\partial t}+u_i{\partial }/{\partial x_i}$
is the material derivation operator,
$\omega _i$
is the vorticity component and
$\epsilon _{\textit{ijk}}$
is the permutation tensor. More precisely, the above equation describes only the evolution of enstrophy in the near field, without establishing a direct connection to far-field variables. However, given the close relationship between the enstrophy and quadrupole sound sources (Li et al. Reference Li, Ye, Wan, Liu, Sun and Lu2020), the source term
$S_1$
is still employed for acoustic diagnostics in the near field.
The second source term
$S_2=\partial (\epsilon _{\textit{ijk}}\omega _i u_j)/{\partial x_k}$
is contained in the wave equation for total enthalpy (Howe Reference Howe1978, Reference Howe1998). If the viscous effect is neglected at a low Mach number, the equation reads
\begin{equation} \left \{ D\left(\frac {1}{c^2}D\right)-\frac {1}{\rho }\boldsymbol{\nabla }\boldsymbol{\cdot }(\rho \boldsymbol{\nabla }) \right \} B = \underbrace {\boldsymbol{\nabla }\boldsymbol{\cdot }(\boldsymbol{\omega } \times \boldsymbol{u})}_{S_2\text{:vortex-sound term}} \qquad \underbrace {- \boldsymbol{\nabla }\boldsymbol{\cdot }(T\boldsymbol{\nabla }s)}_{\text{enstrophy source}} + \underbrace {\frac {\partial }{\partial t}\left(\frac {K T}{C_p}Ds\right)}_{\text{unsteady heating}}, \end{equation}
where
$B=C_p T+1/2u_iu_i$
is the total enthalpy,
$K=-(1/\rho )(\partial \rho /\partial T)_p$
is the coefficient of expansion and
$C_p$
denotes the specific heat at constant pressure.
The third source term
$S_3={ ( {(\partial u_i}/{\partial x_j})({\partial u_j}/{\partial x_i})-({\partial u_i}/{\partial x_i})({\partial u_i}/{\partial x_i} ))}$
, or the longitudinal field source term, is provided by the general wave equation for the thermodynamic variable
$\mathcal{R}=\textit{ln}(\rho /\rho _\infty )$
(Mao et al. Reference Mao, Kang, Wu, Yu, Gao, Su and Lu2020), yielding
\begin{equation} \begin{aligned} D^2 \mathcal{R}-(D\mathcal{R})^2-\boldsymbol{\nabla }\boldsymbol{\cdot }(c^2\boldsymbol{\nabla }\mathcal{R})-\nu _\theta \boldsymbol{\nabla} ^2 D\mathcal{R} = \underbrace {\frac {\partial u_i}{\partial x_j}\frac {\partial u_j}{\partial x_i}-\left (\frac {\partial u_i}{\partial x_i}\right )^2}_{S_3\text{:pseudo-inviscid work rate}} + \underbrace {\boldsymbol{\nabla }\boldsymbol{\cdot }\big(c^2 \boldsymbol{\nabla }s^*\big)}_{\text{enstrophy source}},\\ \end{aligned} \end{equation}
where
$\nu _\theta =(\lambda +2\mu )/\rho$
is the kinematic longitudinal viscosity,
$\lambda =-2/3\mu$
is the second dynamic viscosity,
$\nu =\mu /\rho$
is the transverse viscosity and
$s^*=(s-s_0)/C_p$
denotes the normalised entropy. The term
$S_3$
, also a measure of the pseudo-‘inviscid work rate’, denotes a source of the longitudinal field. Generally,
$S_3$
takes a positive value at locations dominated by rotation and a negative value at shear-dominated locations.
2.5. The computational cases and validation
The implicit large eddy simulation (Dairay et al. Reference Dairay, Lamballais, Laizet and Vassilicos2017) of a NACA-0012 airfoil was performed at zero incidence angle. The free-stream Mach number is
$\textit{Ma}=0.25$
and the Reynolds number is
$\textit{Re}=4.08\times 10^5$
based on the airfoil chord length
$L$
and the free-stream velocity
$U_\infty$
. To facilitate a direct numerical validation and comparison with the experimental data, a case with a Mach number of
$\textit{Ma}=0.115$
is also considered. The origin of the coordinate system is located at the leading edge of the airfoil, with the
$x$
-axis pointing downstream in the flow direction, the
$y$
-axis perpendicular to the chord line of the airfoil and the
$z$
-axis extending normal to the airfoil plane. The spanwise extent is
$L_z = 0.12 L$
. To promote boundary-layer transition on the wing surface, suction and blowing perturbations are introduced near the leading edge (Wolf, Azevedo & Lele Reference Wang, Hao, Lu, Tong and Song2012). Specifically, a constant suction with an intensity of
$|v_n|=0.02U_\infty$
is imposed within the region
$0.15 \lt x/L \lt 0.175$
and
$0.01 \lt z/L \lt 0.11$
, while blowing of equivalent strength is applied in the adjacent region
$0.175 \lt x/L \lt 0.2$
with identical
$z$
-coordinate bounds. To achieve effective noise reduction, three types of TE are designed: an impedance trailing edge (ITE), an extended trailing edge (ETE) and an extended impedance trailing edge (EITE), as shown in figure 1. It is noted that a cusped TE would offer an alternative approach that may deliver comparable performance to the extended designs. However, the present ETE and EITE configurations provide greater generality, as they can be readily applied to airfoils with a finite incidence angle or cambered geometries. Hence, the extended designs are adopted in the present study.
Sketch of a NACA-0012 airfoil with (a) an ITE and (b) an EITE. The boundary layer is tripped by a constant suction and blowing technique at
$x{\kern-1pt}/{\kern-1pt}L{\kern-1pt}\in [0.15, 0.20]$
and
$z/L\in [0.01, 0.11]$
.
Computational cases and control parameters for the numerical simulations. Here,
$L_p$
is the streamwise extent for the controlled region,
$\Delta t$
is the time step for integration,
$t_s$
is the total time for turbulence statistics and the resistance
$R$
, damping ratio
$\zeta$
and resonance frequency
$f_r$
are parameters for the impedance boundary conditions.
The ITE involves replacing the rear 20 % of the airfoil with an impedance wall, which yields the streamwise extent of
$L_p=0.2L$
. Teruna et al. (Reference Teruna, Avallone, Ragni and Casalino2021b
) has shown that effective noise reduction is difficult to achieve when the flow from the suction and pressure sides lacks interaction. Nevertheless, a complete replacement of the TE section with porous material would adversely affect the structural integrity, and we still investigate the potential impact of introducing separate impedance walls on the upper and lower surfaces. The extended configurations share the same extended length of
$L_p=0.12L$
, consistent with our previous work (Hu et al. Reference Hu, Wan, Sun and Lu2024). Moreover, the EITE configuration incorporates a zero-thickness impedance splitter plate (Zhou et al. Reference Zhou, Koh, Gauger, Meinke and Schöder2025) attached to the TE, which can delay potential vortex shedding induced by TE bluntness. Although the zero-thickness plates are physically unrealisable, the effects of such plates on the flow field are known to be nearly equivalent to those with a small thickness of the order of
$h^+ \sim \textit{O}(1)$
(Jones & Sandberg Reference Jones and Sandberg2012) (
$h^+$
is the plate thickness in wall scales) for high Reynolds number turbulent flow. Scalo et al. (Reference Sano, Abreu, Cavalieri and Wolf2015) indicate that, when the acoustic resistance
$R$
is equal to
$10^0$
, the influence of the impedance wall on the flow is largely negligible. In contrast, the near-wall flow structures undergo a significant change as
$R$
approaches
$10^{-2}$
. Consequently, we allow the acoustic resistance to transition smoothly from
$10^0$
to
$10^{-2}$
across the impedance surface following an exponential decay. The characteristic frequency is fixed for a single case, and it ranges from 4 to 64 for different cases (the fundamental frequency
$f_r=4$
is obtained through a priori resolvent analysis (Yeh & Taira Reference Xiao, Song, He, Liu and Zhou2019)). The distance of the first near-wall grid to the wall,
$\Delta y^+_w$
, is less than 1 in wall scales, ensuring that the near-wall flow is fully resolved. The total statistical sampling duration is approximately 15–18 flow-through times. The detailed parameters are given in table 1. The simulations were performed on a C-type, body-fitted structured grid (Jones & Sandberg Reference Jones and Sandberg2012), designed to maintain maximum orthogonality. The grid dimensions in the streamwise, normal and spanwise directions are
$4631 \times 309 \times 201$
, respectively, with a total of
$2.852\times 10^8$
cells. The mesh resolution is particularly fine around the airfoil surface, accounting for approximately half of the total streamwise grid points. It is noteworthy that the impedance distribution for both the ITE and EITE cases follows a consistent exponential decay, defined as
\begin{equation} R = 10^{-2(x-x_p)/L_p}, \end{equation}
where
$x_p$
denotes the starting position and
$L_p$
represents the streamwise length of the porous treatment. This formulation ensures that the resistance
$R$
transitions from
$10^0$
at the junction to
$10^{-2}$
at the downstream termination. The variation in specific parameters between these cases arises from their distinct physical configurations. For the ITE cases, the porous treatment is applied to the final 20 % of the airfoil chord (
$x_p=0.8, L_p=0.2$
), following established findings that the TE region is most critical for noise radiation (Teruna et al. Reference Teruna, Avallone, Ragni and Casalino2021b
). In contrast, for the ESt64 case, the porous extension is set to
$L_p=0.12$
(
$1.0 \leqslant x \leqslant 1.12$
). This extension length is consistent with our previous studies (Hu et al. Reference Hu, Wan, Sun and Lu2022), which indicated that further increasing the length may introduce additional physical disturbances into the wake sources. Figure 2(a) illustrates the overall distribution of the grid points, featuring a 10-layer buffer zone in the normal direction and a 20-layer buffer zone (Hu, Morfey & Sandham Reference Hu, Morfey and Sandham2003) near the outlet to suppress potential acoustic wave reflections and unphysical oscillations. Figures 2(b) and 2(c) present local details of the leading and trailing edges, respectively, where refinement is applied in regions of rapid flow variations, such as the turbulent transition zone and the TE region. The resolution of the airfoil surface mesh is presented in figure 2(e), confirming that the condition
$y^+\lt 1$
is maintained in the range of
$0.4\leqslant x/L\leqslant 1$
.
Details of the mesh topology for turbulent flow over a NACA-0012 airfoil: (a) overview of the grid distribution (every 5th grid point shown), (b) enlarged view near the leading edge, (c) enlarged view near the TE and (d) mesh resolution along the airfoil surface. Different areas in (
$a\sim c$
) are distinguished by colour: — sponge zone, 
interior region, 
suction region, 
blowing region.
Figure 2. Long description
Panel A: A grid distribution plot showing the mesh topology for turbulent flow over a NACA-0012 airfoil. The plot displays every 5th grid point. The x-axis is labeled x/L and the y-axis is labeled y/L. Different areas are distinguished by color: sponge zone, interior region, suction region, and blowing region. Panel B: An enlarged view near the leading edge of the airfoil. The x-axis is labeled x/L and the y-axis is labeled y/L. The plot shows a detailed grid distribution in this region. Panel C: An enlarged view near the trailing edge of the airfoil. The x-axis is labeled x/L and the y-axis is labeled y/L. The plot shows a detailed grid distribution in this region. Panel D: A line graph showing the mesh resolution along the airfoil surface. The x-axis is labeled x/L. The y-axis on the left is labeled with Δx+, Δz+, and the y-axis on the right is labeled with Δy+. The graph includes three lines representing Δx+, Δz+, and Δy+ respectively, with different colors for each line.
Numerical validation of the averaged kinematic field for the case M0115S: (a) pressure coefficient
$C_p$
,(b) wall skin-friction coefficient
$C_{{\kern-1pt}f}$
, (c) tangential velocity at
$x/L\simeq 0.85$
, (d) strength of the velocity fluctuation at
$x/L\simeq 0.85$
, (e) wake velocity profile at
$x/L\simeq 1.05$
and (f) streamwise velocity fluctuation strength at
$x/L\simeq 1.05$
, with reference to previous numerical (Wolf et al. Reference Wang, Hao, Lu, Tong and Song2012; Ribeiro et al. Reference Powell2016; Bodling & Sharma Reference Bodling and Sharma2019) and experimental results (Garcia-Sagrado & Hynes Reference Garcia-Sagrado and Hynes2012).
The results of case M0115S are used to validate the numerical results, consistent with the parameters set by previous studies (Brooks, Pope & Marcolini Reference Brooks, Pope and Marcolini1989; Wolf et al. Reference Wang, Hao, Lu, Tong and Song2012; Garcia-Sagrado & Hynes Reference Garcia-Sagrado and Hynes2012), as shown in figure 3. Figure 3(a) compares the averaged pressure coefficient
$C_p = (p-p_{\infty })/(({1}/{2})\rho _\infty U_\infty ^2)$
along the airfoil surface, showing excellent agreement except for the tripping region. The distribution of friction coefficient
$C_f=\tau _w/(({1}/{2})\rho _\infty U_\infty ^2)$
is shown in figure 3(b), where
$\tau _w$
is the wall shear stress. Near the tripping region, the friction coefficient increases sharply and then decreases towards the TE. Moreover, good consistency can also be seen in figures 3(c) and 3(d), which compare the time-averaged tangential velocity and velocity fluctuations. Here, the profiles are extracted at
$x/L = 0.85$
. In figure 3(e), the wake velocity profile is in better agreement with the numerical simulation results. Still, there is a certain discrepancy with the experimental results at the valley position, which may be due to different turbulent inflow conditions. The wake velocity fluctuations in figure 3(f) further indicate that the differences between the present results and those of previous studies are minor. Overall, the present numerical results demonstrate high reliability and fidelity in kinematic statistics.
Figure 4(a) presents the wall-pressure power spectrum density,
$\phi(\omega)$
, at
$x/L \simeq 0.91$
in wall viscous scales for the case M0115S. For
$\omega \nu /u_\tau ^2 \lesssim 0.5$
, the spectral levels remain largely unchanged. Within the higher-frequency range, the spectrum exhibits a two-stage decay, first following an approximately
$\omega ^{-5}$
power law and subsequently an
$\omega ^{-8}$
decay. The wall spectrum demonstrates strong overall agreement with the findings of Wolf et al. (Reference Wang, Hao, Lu, Tong and Song2012), although small deviations are present at higher frequencies. To facilitate a direct comparison of the far-field sound pressure with existing experimental results, the following scaling method (Herr & Reichenberger Reference Herr and Reichenberger2011; Moreau & Doolan Reference Moreau and Doolan2013) is adopted:
\begin{equation} \begin{aligned}\textit{SPL}_{1/3,scaled} &= \textit{SPL}_{1/3} - 50\textit{log}(\textit{Ma})-10\textit{log}(\delta L_z/R_s^2),\\ {\textit{St}}_\delta &= f\delta /U_\infty , \end{aligned} \end{equation}
where
$\textit{SPL}_{1/3}$
is the one-third octave band spectrum,
$\delta$
is the characteristic outer length scale (such as the turbulent boundary-layer thickness or the displacement thickness at the TE) and
$R_s$
is the distance from the TE to the monitoring point. The spectrum results in figure 4(b), sampled at a far-field point directly above the TE, exhibit distinct peaks at
${\textit{St}}_\delta \simeq 0.25$
. As the frequency continues to increase, the spectral energy gradually decays, maintaining a high level of accuracy with a maximum deviation of less than 2 dB from the experimental reference.
Numerical validation of acoustic variables for the case M0115S: (a) comparison of the wall-pressure power spectral density (PSD) at
$x/L\simeq 0.91$
with the numerical reference data from Wolf et al. (Reference Wang, Hao, Lu, Tong and Song2012), and (b) comparison of the normalised one-third octave band spectrum with the experimental results by Brooks et al. (Reference Brooks, Pope and Marcolini1989), sampled at a distance of
$R_s=8L$
above the TE.
3. Results and discussion
3.1. Noise reduction by impedance trailing edges
To investigate the effectiveness of various designs, figure 5 compares the noise reduction performance of solid wall and impedance TE cases in overall sound pressure level (OSPL). Previous studies (Howe Reference Howe1979; Schulze & Sesterhenn Reference Schmidt and Towne2013; Wang et al. Reference Vinuesa, Negi, Atzori, Hanifi, Henningson and Schlatter2021) have suggested that the streamwise-varied impedance is crucial for the suppression of TE radiated noise. However, the current results demonstrate that a pure ITE (configuration I in figure 1) does not necessarily lead to noise reduction and may even cause noise to increase. Specifically, at a lower characteristic frequency (
$f_r=4$
), the pure impedance wall significantly enhances noise radiation. This noise enhancement effect progressively diminishes as the characteristic frequency increases, approaching the noise radiation level of a traditional solid TE at
$f_r =64$
. This finding further highlights the limitations of the sound absorption capabilities of a pure ITE when there is no flow interaction between the airfoil’s upper and lower surfaces (Teruna et al. Reference Teruna, Avallone, Ragni, Rubio-Carpio and Casalino2020, Reference Teruna, Avallone, Ragni and Casalino2021b
). To address this, another tailored design, specifically an extension plate, is thoroughly explored. In fact, a key distinction between an airfoil and a flat plate is the finite thickness, or the wedge angle (Spiropoulos et al. Reference Skote and Henningson2025), of the former’s TE. This leads to an abrupt change in the flow direction on both the upper and lower surfaces to satisfy the Kutta condition. Hence, a natural approach is to reduce the TE thickness, as a flat-plate extension inherently satisfies the Kutta condition (Amiet Reference Amiet1990). This strategy has been validated through numerical simulations, where the additional solid plate (ETE) achieves noise reduction of 3.62 dB in the direction of maximum noise radiation. When the additional impedance plate (EITE) (configuration II in figure 1) is adopted, the noise reduction can be further increased to 4.22 dB. Previously, an additional solid flat plate was adopted by Jones & Sandberg (Reference Jones and Sandberg2012) as a baseline for validating the effectiveness of the serrated TE. However, no direct comparison was made with a traditional airfoil. It is also worth noting that the ETE in the present study is twice the length of that used by Jones & Sandberg (Reference Jones and Sandberg2012), which could be a potential mechanism affecting the noise reduction capability.
Acoustic performance of the airfoil with solid or impedance TEs: (a) the directivity of the OSPL, (b) the change in noise levels for different cases, where the negative values represent noise reduction. 
Figure 5. Long description
Panel A: A polar plot shows the directivity of the overall sound pressure level (OSPL) for different airfoil configurations. The plot uses radial coordinates with angles labeled in radians and degrees, and concentric circles indicating OSPL values. Multiple lines represent different cases, including M025S, St04, St08, St16, St32, St64, Esolid, and ESt64. Panel B: A line graph displays the change in noise levels (ΔOSPL/dB) as a function of the angle θ (in radians). The horizontal axis ranges from 0 to 2π, and the vertical axis ranges from -6 to 30 dB. Multiple lines represent different cases, with negative values indicating noise reduction.
Directivity plots of the sound power, averaged on a circle with radius
$R_s=4L$
about the TE, shown in different frequency bands: (a) low-frequency range
$1\lt \textit{St}\lesssim 3$
, (b) mid-frequency range
$3\lt \textit{St}\lesssim 5$
and (c) high-frequency range
$5\lt \textit{St}\lesssim 10$
. The sound power is normalised with respect to the circumferential average of the reference case for each frequency band. For clarity of comparison, the results from some of the noise-increasing cases are not fully displayed where they extend beyond the plotting domain. 
Figure 6 further compares the noise directivity distribution at a distance of
$R_s=4L$
from the TE across different frequency domains, where the frequency bands are distinguished based on their distinct spectral decay characteristics. In the low-frequency range of
$1\lt \textit{St}\lesssim 3$
(the Strouhal number
$\textit{St}=fL/U_\infty$
is the non-dimensionalised frequency based on the airfoil chord length and far-field free-stream velocity), noise radiation exhibits typical dipole characteristics without obvious change in the noise radiation direction (Jiang et al. Reference Jiang, Moreau, de Silva and Doolan2024). Significant differences are observed between the alternative designs for configuration I, all of which exhibit noise enhancement. In contrast, the ETE/EITE of configuration II demonstrates a substantial reduction in sound radiation levels. As frequency increases, the non-compact nature of the sound source begins to dominate when the frequency falls in
$3\lt \textit{St}\lesssim 5$
, manifested as two main lobe peaks in the upper or lower half-plane (Avallone et al. Reference Avallone, Van Der Velden, Ragni and Casalino2018; Teruna et al. Reference Teruna, Avallone, Ragni, Rubio-Carpio and Casalino2020), and the extended designs show differences in this frequency range. At higher frequencies of
$5\lt \textit{St}\lesssim 10$
, the circumferential distribution of radiated noise becomes irregular. This frequency-dependent behaviour aligns with the observations of Cavalieri, Wolf & Jaworski (Reference Cavalieri, Wolf and Jaworski2016) for finite perforated elastic TEs, further validating that the current numerical approach effectively captures the fundamental scattering physics associated with TE treatments. Nevertheless, the primary radiation direction remains inclined upstream, and the noise reduction trend is consistent with that observed at lower frequencies. Consequently, the alternative impedance design of configuration I (ITE) exhibits an increase in noise radiation levels across the entire frequency domain, while the extended design of configuration II (ETE/EITE) shows the opposite trend, demonstrating excellent noise reduction performance. Furthermore, the impact of EITE design is primarily more pronounced in the mid-frequency range, compared with the ETE.
3.2. Flow structures and turbulence statistics
This subsection focuses on the correlation between TE noise radiation and the near-field flow structure. Figure 7 illustrates the vortex structure visualised using Q-criterion iso-surfaces (
$Q=-((1/2)(\partial u_{i}/\partial x_{j})(\partial u_{j}/\partial x_{i}))$
) and dilatation field distribution in the near field. It is evident that constant suction/blowing effectively promotes a rapid transition in the flow over the wing surface, ensuring a turbulent incoming flow over the TE (Wolf et al. Reference Wang, Hao, Lu, Tong and Song2012; Sano et al. Reference Rienstra and Hirschberg2019; Bodling & Sharma Reference Bodling and Sharma2019). Near the TE, significant sound radiation is observed, with the radiation centre moving downstream for the extended plate (Hu et al. Reference Hu, Wan, Sun and Lu2024). Simultaneously, the introduction of ITE leads to the reappearance of vortex shedding at the TE, particularly in cases with lower characteristic frequencies, as depicted in figure 7(
$b{-} d$
). This vortex shedding contributes to a significant intensification of the radiated noise (see also figure 5). As the characteristic frequency increases, vortex shedding gradually attenuates and becomes scarcely discernible when
$f_r$
approaches 64. In other words, to avoid resonance, the characteristic frequency
$f_r$
should be selected sufficiently far from the most amplified perturbation frequency of the baseline flow.
Iso-contours of vortex structures coloured by streamwise velocity and slice view of the dilatation. Cases: (a) M025S, (b) St04, (c) St08, (d) St16, (e) St32, (f) St64, (g) Esolid, (h) ESt64. For visual clarity, labels are provided only in subfigures (a) and (b). However, the red arrow, orange box and dashed green lines consistently denote the regions for noise radiation centre, periodic vortex shedding and propagating acoustic waves throughout the figure.
Figure 7. Long description
Panel A: An image showing vortex structures colored by streamwise velocity. The red arrow indicates the noise radiation center. Panel B: An image showing vortex structures colored by streamwise velocity. The red arrow indicates intensified sound radiation, the orange box highlights vortex shedding, and the dashed green lines denote propagating acoustic waves. Panel C: An image showing vortex structures colored by streamwise velocity. The red arrow indicates the noise radiation center. Panel D: An image showing vortex structures colored by streamwise velocity. The red arrow indicates the noise radiation center. Panel E: An image showing vortex structures colored by streamwise velocity. The red arrow indicates the noise radiation center. Panel F: An image showing vortex structures colored by streamwise velocity. The red arrow indicates the noise radiation center. Panel G: An image showing vortex structures colored by streamwise velocity. The red arrow indicates the noise radiation center. Panel H: An image showing vortex structures colored by streamwise velocity. The red arrow indicates the noise radiation center.
Slice views of the wall-tangential velocity
$u_t$
, wall-normal velocity
$v_n$
and pressure
$p$
(from left to right in each subfigure) for the following cases:(a) M025S, (b) St04, (c) St08, (d) St16, (e) St32, (f) St64, (g) Esolid, (h) ESt64. The regions of adverse pressure gradient (APG) and FPG are also marked.
To facilitate a comparative analysis of the evolution of near-field flow structures across different cases, figure 8 illustrates the variations in primitive variables. The streamwise velocity
$u_t$
is sampled within the viscous layer, while the normal velocity
$v_n$
and pressure
$p$
are sampled on the wall surface. A comparison of figure 8(a–f) shows that the impedance wall disrupts the streamwise velocity streaks near the TE of the airfoil and induces intense vortex shedding (see also in figure 7). At the low characteristic frequency
$f_r=4$
, the spanwise rollers, driven by Kelvin–Helmholtz instability (Scalo et al. Reference Sano, Abreu, Cavalieri and Wolf2015; Chen & Scalo Reference Chen and Scalo2021b
), span almost the entire computational domain in the spanwise direction. As the characteristic frequency increases, the streamwise extent of the TE vortices gradually diminishes, becoming barely noticeable when
$f_r = 64$
. In figure 8(g), typical streamwise velocity streaks are also present on the extended flat plate in case Esolid, and the velocity magnitude is markedly higher as a result of the favourable pressure gradient (FPG) (Balin & Jansen Reference Balin and Jansen2021). Figure 8(h) presents the flow structures for the EITE case ESt64. Compared with the ETE case, the velocity and pressure distributions show only minor changes. Therefore, the impedance wall primarily modifies the velocity and pressure distributions in the immediate vicinity of the TE rather than the upstream flow (
$x \lesssim 0.8$
for ITE cases and
$x \lesssim 1$
for extended cases). Furthermore, for ITE cases, this localised influence progressively diminishes as the characteristic frequency increases, indicating a weakened coupling between the impedance boundary and the high-frequency fluctuations.
Comparison of turbulent statistics along the streamwise direction: (a) the pressure coefficient
$C_p$
, (b) the friction coefficient
$C_{{\kern-1pt}f}$
, (c) the turbulent boundary-layer displacement thickness
$\delta ^*$
and momentum thickness
$\theta$
, (d) the wall-penetration coefficient
$C_\eta = \overline {\rho } \widetilde {v^{\prime \prime } v^{\prime \prime }}/Q$
and (e) the root mean square of wall-pressure jump
$(p_{ss}-p_{ps})_{\textit{rms}}/Q$
(
$p_{ss}$
is the wall pressure on the suction side, and
$p_{ps}$
is the wall pressure on the pressure side), where the light-black and light-red shaded regions denote the areas covering the final 10 % chord length of the airfoil. 
Figure 9. Long description
Five line graphs compare turbulent statistics along the streamwise direction. Panel A: The line graph shows the pressure coefficient (C_p) on the y-axis and the streamwise direction (x/L) on the x-axis. Panel B: The line graph shows the friction coefficient (C_f) on the y-axis and the streamwise direction (x/L) on the x-axis. Panel C: The line graph shows the turbulent boundary-layer displacement thickness (delta asterisk) and momentum thickness (theta) on the y-axis and the streamwise direction (x/L) on the x-axis. Panel D: The line graph shows the wall-penetration coefficient (C_eta) on the y-axis and the streamwise direction (x/L) on the x-axis. Panel E: The line graph shows the root mean square of wall-pressure jump ((p_ss - p_ps)_rms / Q) on the y-axis and the streamwise direction (x/L) on the x-axis. The light-black and light-red shaded regions denote the areas covering the final 10 percent chord length of the airfoil.
In the following sections, we will concentrate our analysis on the St64, Esolid and ESt64 cases to enable a direct comparison. Figure 9 illustrates the spatial distribution of different turbulent statistics along the streamwise direction. In figure 9(a), the pressure coefficient
$C_p = (p-p_\infty )/Q$
(
$Q=1/2\rho _\infty U_\infty ^2$
is the far-field dynamic pressure) is similar across different cases for
$x/L\lt 1$
, in agreement with the findings by Teruna et al. (Reference Tam and Auriault2021a
). After the position of
$x/L=1$
, the pressure for extended cases decreases rapidly, suggesting an abrupt change from an APG to an FPG. The distribution of the wall skin-friction coefficient
$C_f = \tau _w/Q$
is markedly different, as shown in figure 9(b). The
$C_{{\kern-1pt}f}$
for impedance case St64 is slightly higher than that in the baseline case, while cases Esolid and ESt64 exhibit a sudden increase followed by an approximate plateau for
$x/L\gt 1$
. The increase in
$C_{{\kern-1pt}f}$
indicates that the significant noise reduction achieved by the extended plate design is accompanied by a deficit in aerodynamic performance, which is also reported by previous studies (Koh, Schröder & Meinke Reference Koh, Schröder and Meinke2018; Teruna et al. Reference Teruna, Avallone, Ragni, Rubio-Carpio and Casalino2020). In figure 9(c), the trend of boundary-layer displacement thickness
$\delta ^*$
or momentum thickness
$\theta$
closely follows a trend similar to the pressure coefficient, suggesting the significant impact of the pressure gradient on airfoil surface turbulence (Vinuesa et al. Reference Towne, Schmidt and Colonius2018). Specifically, an APG promotes the growth of the turbulent boundary layer, whilst an FPG results in rapid thinning. The penetration coefficient
$C_\eta = \overline {\rho } \widetilde {v^{\prime \prime } v^{\prime \prime }}/Q$
in figure 9(d), or the normalised Reynolds stress, is strongly correlated with the extra noise generation of dipole source on the wall surface, as indicated by the Ffowcs Williams–Hawkings equation (Ffowcs Williams & Hawkings Reference Ffowcs Williams and Hawkings1969). It can be observed that the extended impedance configuration ESt64 yields a comparatively modest increase in
$C_\eta$
attributed to alterations in the TE flow structures. In contrast, the pure ITE induces a far more substantial amplification of the coefficient
$C_\eta$
for case St64. Figure 9(e) further illustrates the root mean square of the wall-pressure jump along the streamwise direction. For both the solid and alternative impedance walls, the wall-pressure jump gradually increases as it approaches the TE, where it exhibits a typical spike before rapidly decaying (Ayton et al. Reference Ayton, Gill and Peake2016). In contrast, the maximum values for the extended cases are observed at
$x/L \simeq 1$
, after which, the wall-pressure jump decays rapidly and is significantly lower than the value of the baseline case at
$x/L \simeq 1$
. Consequently, the extended plate geometry results in a smoother satisfaction of the Kutta condition at the TE, leading to a potentially weaker acoustic scattering.
Profiles of the averaged velocity, velocity fluctuation and pressure fluctuation in inner scales sampled at
$x/L$
= 0.75, 0.85, 0.95 and 1.07 from bottom to top: (a) wall-tangential-averaged velocity
$\widetilde u_t^+$
. The adjacent lines have a vertical offset of 25. For the solid case M025S and the impedance case St64, the profile originally at a position of 0.95 is copied to 1.07 to facilitate a clear comparison, (b) wall-tangential velocity fluctuation
$(u_t^{\prime \prime })_{\textit{rms}}^+$
profiles with a normal offset of 3.0, (c) wall-normal velocity fluctuation
$(v_n^{\prime \prime })_{\textit{rms}}^+$
profiles with a normal offset of 1.5, (d)
$-(\widetilde {u_t^{\prime \prime } v_n^{\prime \prime }})^+$
profiles with a normal offset of 1.5, (e)
$(w^{\prime \prime })_{\textit{rms}}^+$
profiles with a normal offset of 1.5, (f)
$(p^{\prime })_{\textit{rms}}^+$
profiles with a normal offset of 5.0. 
Figure 10. Long description
The image contains six line graphs labeled (a) through (f), each depicting different profiles of averaged velocity, velocity fluctuation, and pressure fluctuation in inner scales at various positions. Panel A: A line graph showing wall-tangential-averaged velocity profiles. The x-axis is labeled y+ and the y-axis is labeled u+_t. The adjacent lines have a vertical offset of 25. For the solid case M025S and the impedance case St64, the profile originally at a position of 0.95 is copied to 1.07 to facilitate a clear comparison. Panel B: A line graph showing wall-tangential velocity fluctuation profiles with a normal offset of 3.0. The x-axis is labeled y+ and the y-axis is labeled (u’_t)_rms+. Panel C: A line graph showing wall-normal velocity fluctuation profiles with a normal offset of 1.5. The x-axis is labeled y+ and the y-axis is labeled (v’_n)_rms+. Panel D: A line graph showing profiles with a normal offset of 1.5. The x-axis is labeled y+ and the y-axis is labeled -(u’_t v’_n)_rms+. Panel E: A line graph showing profiles with a normal offset of 1.5. The x-axis is labeled y+ and the y-axis is labeled (w’_t)_rms+. Panel F: A line graph showing profiles with a normal offset of 5.0. The x-axis is labeled y+ and the y-axis is labeled (p’)_rms+. Labels: M025S; St64; Esolid; ESt64.
The close correlation between acoustic emissions and near-field flow unsteadiness (Blake Reference Blake1986) prompts an investigation into the spatial distribution of fundamental turbulence statistics in the boundary layer, as presented in figure 10. The averaged wall-tangential velocity profile in figure 10(a) exhibits excellent agreement with the theoretical prediction of the wall within the viscous sublayer. However, a deviation from the classical log law is observed within the logarithmic region, attributed to the effects of APG (Skote & Henningson Reference Sieber, Paschereit and Oberleithner2002). The discrepancies between the different cases are minimal at the streamwise locations of
$x/L \simeq 0.75$
and
$x/L \simeq 0.85$
, and a pronounced downward shift in the velocity profile is evident for the impedance case at
$x/L \simeq 0.95$
, which is primarily due to an increase in the wall skin friction induced by the impedance boundary condition (Scalo et al. Reference Sano, Abreu, Cavalieri and Wolf2015; Chen & Scalo Reference Chen and Scalo2021b
). The downward shift of the profile is more pronounced at the streamwise position
$x/L \simeq 1.07$
(the distance from this point to the TE measures 0.05
$L$
), consistent with the increase in the wall friction coefficient (see figure 9(b)). Figure 10(b) presents the distribution of the wall-tangential velocity fluctuation, revealing a reduction in these fluctuations near the TE for all controlled cases. The extended impedance case (ESt64) exhibits a distinct bimodal profile at
$x/L\simeq 1.07$
, with peaks located within the buffer layer at
$y^+ \simeq 15$
and at a higher position of
$y^+\simeq 200$
, which suggests a significant modification of the flow structures within the boundary layer by the FPG. In figure 10(c), case St64 exhibits a significantly elevated fluctuation intensity in the near-wall region. Although the overall distributions for cases Esolid and ESt64 are remarkably similar, the peak in normal velocity fluctuations for these cases is shifted away from the wall, coinciding approximately with the position of the secondary peak observed in the streamwise velocity fluctuations. Changes in shear stress are subsequently explored in figure 10(d). It is shown that the ITE case St64 exhibits a greater intensity near the wall (Scalo et al. Reference Sano, Abreu, Cavalieri and Wolf2015; Chen & Scalo Reference Chen and Scalo2021a
) but a reduction at the peak location. Furthermore, the location of this peak is also shifted upwards to
$y^+\simeq 200$
. The trend is also observable in the spanwise velocity fluctuations in figure 10(e). Figure 10(f) presents the changes in wall-pressure fluctuations. Compared with the baseline case, case St64 results in a modest increase in fluctuation intensity at
$x/L \simeq 0.95$
. Conversely, the ETE/EITE causes a substantial reduction in pressure intensity at
$x/L \simeq 1.07$
. This finding, which is consistent with the noise reduction effects discussed previously, implies the dominant role of wall dipole sources in the acoustic radiation process. In short, the extended plate attenuates the streamwise and spanwise Reynolds stresses, as well as the pressure fluctuations in inner scales, within the turbulent boundary layer on its surface, with minimal difference observed between ETE and EITE configurations.
3.3. Dynamic evolution of flow structures
As previously demonstrated, the impedance wall was found to have a significant effect on near-field flow structures. To provide a deeper insight into the dynamic evolution of the spatial-temporal coherence structures near the impedance TEs, the spectral proper orthogonal decomposition (SPOD) algorithm (Sieber, Paschereit & Oberleithner Reference Shubham, Sandberg, Moreau and Wu2016; Schmidt & Towne Reference Scalo, Bodart and Lele2017; Towne, Schmidt & Colonius Reference Teruna, Manegar, Avallone, Ragni, Casalino and Carolus2018) is employed. The SPOD method has proven effective in analysing the frequency-specific energetic structures for turbulence over the airfoil (Sano et al. Reference Rienstra and Hirschberg2019; Yuan et al. Reference Yeh and Taira2025), as it combines the aspects of proper orthogonal decomposition (Berkooz, Holmes & Lumley Reference Berkooz, Holmes and Lumley1993) and spectral analysis. The related algorithm is briefly introduced here.
Considering a time series of snapshot data
$p^\prime _i = p^\prime (\boldsymbol{x}, t_i)(i=1,2,\ldots ,N)$
, where
$\boldsymbol{x}$
is the spatial coordinate,
$t_i$
is the discrete time instant. The time-series data matrix
$Q= [p^\prime _1, p^\prime _2,\ldots ,p^\prime _N ]$
is first segmented into
$N_{blk}$
overlapping blocks. For data block
$n$
, the data matrix
$Q^n$
is then assembled as
\begin{equation} Q^n = \left [ p^\prime _{1+(n-1)(N_{\!f}-N_o)},p^\prime _{2+(n-1)(N_{\!f}-N_o)},\ldots ,p^\prime _{N_{\!f}+(n-1)(N_{\!f}-N_o)} \right ], \end{equation}
where
$N_{\!f}$
is the segment length for the Fourier transform, and
$N_o$
is the number of overlapped points. Next, a windowed discrete Fourier transform (DFT) is adopted to transform the data from the time domain to the frequency domain
\begin{equation} \hat {Q}^n = \textit{DFT}(Q^n)=\left [\hat {\boldsymbol{p}}_1^n,\hat {\boldsymbol{p}}_2^n,\ldots , \hat {\boldsymbol{p}}_{N_{\!f}}^n\right ], \end{equation}
where the column
${\boldsymbol{p}}_k^n$
represents the
$n$
th realisation of the Fourier mode at the discrete frequency
$f_k$
.
For each frequency
$f_k$
, we collect the
$k$
th column of each block
$\hat {Q}^n$
to form a new matrix
\begin{equation} \hat Q_{f_k} = \sqrt {\Delta t/\left(N_{blk} \sum _{j=1}^{N_{\!f}}w_j^2\right)}\left [{\boldsymbol{p}}_k^1,\hat {\boldsymbol{p}}_k^2,\ldots , \hat {\boldsymbol{p}}_{k}^{N_{blk}}\right ], \end{equation}
where
$\Delta t$
is the time interval,
$w_j$
is the component of the discretised windowing function. The cross-spectral density matrix is then constructed as
\begin{equation} M_{f_k} = \hat Q_{f_k}^* W \hat Q_{f_k}, \end{equation}
where
$W$
is the weighting matrix, and the
$N_{blk}\times N_{blk}$
eigenvalue decomposition problem is formulated by
\begin{equation} M_{f_k} \varTheta _{f_k} = \varTheta _{f_k} \widetilde {\varLambda }_{f_k}. \end{equation}
Finally, the SPOD modes
$\widetilde {\varPsi }_{f_k}=\hat {Q}_{f_k} \varTheta _{f_k} \widetilde {\varLambda }_{f_k}^{-1/2}$
and the modal energy
$\widetilde {\varLambda }_{f_k}$
for the
$k$
th discrete frequency are obtained.
Comparison of the energy of the first four SPOD modes for different cases: (a) the first mode, (b) the second mode, (c) the third mode and (d) the fourth mode. 
Given the spanwise periodicity, the SPOD analysis is performed using the pressure fluctuations observed on the
$x{-}y$
plane slice. Figure 11 compares the energy distribution across the first four modes among four typical cases. In figure 11(a), all cases exhibit a significant energy peak around
$\textit{St}\simeq 25$
, and this peak remains unaffected by changes in the TE boundary conditions, suggesting it is inherent to the flow system. Furthermore, the frequency of this peak is considerably higher than that of the far-field noise, which means it predominantly describes the evolution of the near-field flow structures. Beyond this frequency, the spectral energy rapidly drops off. Compared with the baseline case, the modal energy for the St64 case is higher, indicating that it leads to greater near-field hydrodynamic disturbances, while the extended cases (Esolid and ESt64) show slightly lower energy. A similar distribution pattern is observed for the second mode shown in figure 11(b). However, the energy peak is not as pronounced as in the first mode, and the overall modal energy level is lower. For the third and fourth modes (figures 11
c and 11
d), there is no clear energy peak, and the differences between the cases are also smaller. The modal energy remains relatively constant at
$\textit{St} \lesssim 30$
before rapidly decaying thereafter. Consequently, the distribution of modal energy across all cases is broadly similar, with the exception of the St64 case, which shows a moderate increase in modal energy.
Distribution of the first pressure SPOD mode: (a) M025S, (b) St64, (c) Esolid, (d) ESt64 and (e) comparison of the pressure jump between the upper and lower walls.
The pressure distribution of the first SPOD mode is shown in figure 12. For all cases, the coherent structures are concentrated adjacent to the airfoil surface and arise from the location of turbulence transition, where constant blowing and suction are imposed (Sano et al. Reference Rienstra and Hirschberg2019). Due to the effect of the pressure gradient, the scales of coherent structures gradually increase along the airfoil surface (Yuan et al. Reference Yeh and Taira2025). As the flow approaches the TE (
$x/L\to 1$
), the coherent structures over the airfoil surface in cases M025S and St64 undergo a process of breakdown and reorganisation, with evidently higher mode energy near the TEs. For the extended flat-plate case Esolid, the ordered-to-disordered transition of model structures primarily occurs within the interval
$L \lt x \lt L+L_p$
, and is less prominent in case ESt64. The changes in model structure reflect the mechanism by which the coherent structures adapt to the TE. Upon entering the wake region, the coherent structures gradually break down after persisting for a certain distance. Notably, the coherent structures break down at
$x/L\simeq 1.3$
in case St64, whereas in other cases, the breakdown occurs around
$x/L=1.2$
. As the TE in the extended cases is located further downstream, the behaviour clearly indicates that the coherent structures in the extended cases experience more rapid mode fragmentation upon leaving the TE. At positions far from the wall, the anti-symmetric pressure waves propagating forward relative to the
$y=0$
plane can be clearly observed (Sano et al. Reference Rienstra and Hirschberg2019), which is associated with the sound radiation by the wall-pressure jump near the TE (Amiet Reference Amiet1975), as suggested by the TE Kutta condition. However, the modal energy for these acoustic waves is significantly lower than that of the coherent structures near the wall. To facilitate a direct comparison, figure 12(e) illustrates the spatial distribution of the wall-pressure mode jump across the upper and lower surfaces. The phase mismatch between the surfaces results in a clear oscillation in this jump value. Notably, the extended solid or impedance plate exhibits a much smoother, attenuated oscillation as the flow approaches the TE (see also in figure 9(e)), which is expected to reduce acoustic scattering. This observation provides additional evidence for the utility of the Kutta condition in guiding effective TE noise mitigation strategies.
3.4. Wall-pressure jump associated with the Kutta condition
Confined to the inviscid Kutta condition, the velocity or pressure must be continuous at the TE. When considering the unsteady, viscous and turbulent nature of the incoming flow, the scattering field should induce a significant pressure jump across the suction and pressure sides (Amiet Reference Amiet1976; Lyu et al. Reference Lyu, Azarpeyvand and Sinayoko2016) to remove the singularity at the TE. The wall-pressure jump also serves as an important dipole source and is responsible for the generation of the loading noise (Jiang et al. Reference Jiang, Moreau, de Silva and Doolan2024). Previously, Amiet (Reference Amiet1976) derived a formula for the far-field noise spectrum
$S_{pp}$
generated by a semi-infinite plate geometry as
\begin{equation} S_{pp}(x, y, 0, \omega ) = D_\theta |\mathcal{L}|^2 l_z(\omega ) S_{qq}(\omega ) , \end{equation}
where
$\mathcal{L}$
is the gust response function that governs the scattering efficiency,
$l_z(\omega )$
is the spanwise correlation length,
$S_{qq}(\omega )$
is the spectrum for wall-pressure jump and
$D_\theta$
represents the spatial directivity related to the infinitely thin plate geometry and Mach number. This theory suggests that the far-field radiated noise is fundamentally determined by the spanwise correlation length, the wall-pressure spectrum and the response function linked to the edge geometry and the turbulence convection properties. It is worth noting that although the prediction theory was originally derived for flat plates with rigid walls, these key parameters remain crucial for elucidating noise reduction mechanisms subjected to impedance walls (Jiang et al. Reference Jiang, Moreau, de Silva and Doolan2024) and will be examined in detail in the subsequent part.
Comparison of (a) the mean local convection velocity and (b) the spanwise integral length along the streamwise direction. 
As reported by Jaworski & Peake (Reference Jaworski and Peake2013), the TE convection velocity is one of the critical factors that influence the noise radiation efficiency. To facilitate the subsequent analysis, we first define the spatio-temporal correlation function for wall-pressure fluctuation
\begin{equation} C(\Delta x, \Delta z, \Delta t; x, z) = \frac {\langle p^\prime (x, z, t) p^\prime (x+\Delta x, z+\Delta z, t+\Delta t)\rangle }{p^\prime _{\textit{rms}} (x, z) p^\prime _{\textit{rms}} (x+\Delta x, z+\Delta z)}, \end{equation}
where
$\langle \boldsymbol{\cdot }\rangle$
is the ensemble average operator,
$\Delta x$
is the wall-tangential separation,
$\Delta z$
is the spanwise separation and
$\Delta t$
is the temporal lag, respectively. Figure 13(a) provides a quantitative comparison of the streamwise convective velocity
$U_c$
(He, Jin & Yang Reference He, Jin and Yang2017), defined as
$U_c = {\partial (\Delta x)}/{\partial (\Delta t)}$
(
$\Delta x$
corresponds to the spatial separation that yields the maximum spatio-temporal correlation for a given time delay
$\Delta t$
). As the TE is approached, the convection velocity over the airfoil surface decreases rapidly. For extended solid or impedance plates, the transition from an APG to an FPG leads to an increase in the convection velocity of pressure fluctuation for
$x\gt L$
. Owing to the constraints of wall slip and the enhancement of wall shear stress imposed by the impedance boundary (Scalo et al. Reference Sano, Abreu, Cavalieri and Wolf2015; Chen & Scalo Reference Chen and Scalo2021b
), the convection velocity in case ESt64 exhibits a sharp decline as
$x$
moves downstream to 1.12
$L$
, approaching the value of the solid-wall case at
$x \simeq L$
. Furthermore, we investigated the variations in the spanwise integral scale
$l_z$
, calculated via
\begin{equation} l_z(x, z) = \int _{-\infty }^{\infty } C(0, \Delta z, 0; x, z)\,\text{d}(\Delta z). \end{equation}
It is worth noting that the spanwise correlation length is typically obtained by taking the square root of the magnitude-squared coherence of two-point pressure fluctuations and performing a spanwise integral (Avallone et al. Reference Avallone, Van Der Velden, Ragni and Casalino2018), and is a function of both streamwise position and frequency. The simplified calculation method presented here, however, does not involve frequency variations, but this also facilitates a direct comparison (Wolf et al. Reference Wolf, Azevedo and Lele2015). The distribution of the spanwise integral length scale shown in figure 13(b) exhibits a distinctly opposite trend. Specifically, the integral length scale
$l_z$
gradually increases along the airfoil surface, accompanied by a sharp drop near the TE. Compared with the corresponding solid configurations, the effect of the impedance boundary on the integral length scale is generally less pronounced. Thus, the enhanced performance of the extended impedance wall has a greater influence on the convection velocity instead of the spanwise integral length scale along the wall surface.
For the problem of TE noise, the pressure jump across the wall surface is also a critical source of far-field noise. Figure 14 shows the integrated PSD of the wall-pressure jump for different cases, defined by
\begin{equation} \phi (\omega ) = \iint \varPhi _{\overset {\frown }p\overset {\frown }p}(k_x, k_z, \omega )\,\text{d}k_x\,\text{d}k_z, \end{equation}
where
$\varPhi _{\overset {\frown }p \overset {\frown }p}(k_x, k_z, \omega )$
is the three-dimensional wavenumber–frequency PSD, and
$\varPhi _{\overset {\frown }p \overset {\frown }p}(\omega )$
is the one-dimensional integrated frequency spectrum. The streamwise abscissas are
$x{\kern-1pt}/{\kern-1pt}L{\kern-1pt}\in [1, 1.12]$
for extended cases and
$x{\kern-1pt}/{\kern-1pt}L{\kern-1pt}\in [0.8, 1.0]$
for other cases. In comparison with the solid TE, the ITE case St64 exhibits a highly similar spectral distribution, while also displaying a distinct peak near the characteristic frequency of the impedance boundary, corresponding to
$\omega L/U_e\simeq 2\pi \boldsymbol{\cdot }64$
. In the extended cases (Esolid, ESt64), the energy in the low-frequency range is significantly higher than in the solid-wall case M025S, whereas the high-frequency energy decreases to some extent. To assess the general spectral shape, we compared our numerical results against the empirical Goody model (Goody Reference Goody2004), a widely used tool for predicting turbulent wall-pressure spectra. The model successfully captures the overall trend of the numerical results. However, it exhibits noticeable deviations in the spectral levels near the TE, which highlights the limitations of the model when applied to the complex flow physics near a TE or specialised boundary conditions. For spectra scaled using different parameters, figures 14(b) and 14(d) show a better collapse of the low-frequency spectra across all cases, while figure 14(c) indicates consistency in the high-frequency region. Finally, it is important to note that the wall-pressure spectra shown here are not entirely equivalent to the final far-field noise spectra. The relationship is non-trivial: an observed increase in wall-pressure power does not guarantee a proportional rise in far-field noise, as the TE’s efficiency in scattering the hydrodynamic energy into acoustic waves is the critical factor (Teruna et al. Reference Teruna, Avallone, Ragni and Casalino2021b
).
Power spectral density of wall-pressure jump shown in different scales: (a) scaled by pressure scale
$Q_e=1/2\rho _e U_e^2$
and time scale
$L/U_e$
, (b) scaled by wall shear stress
$\tau _w$
and
$L/u_\tau$
, (c) scaled by
$\tau _w$
and
$\delta ^*/U_e$
, and (d) scaled by
$\tau _w$
and
$\nu /u_\tau$
. The thin dashed lines show the spectrum predicted by the Goody model (Goody Reference Goody2004). 
Wavenumber–frequency spectrum 
of the wall-pressure jump for solid and impedance cases: (a) M025S, (b) St64, (c) Esolid and (d) ESt64. The wavenumber–frequency relationship for the surface wave is plotted at a resistance value of
$R=0.1$
, representing the acoustic resistance at the midpoint of the impedance wall. 
resonance frequency and its harmonics.
Figure 15. Long description
Panel A: A heat map showing the wavenumber-frequency spectrum for the case M025S. The x-axis represents the wavenumber (kxδ*) ranging from -6 to 6, and the y-axis represents the frequency (ωδ*/Ue) ranging from 0 to 4. The color scale indicates the magnitude of the wall-pressure jump, with values from -90 to -30. The convective ridge, acoustic ridge, surface wave, and resonance frequency and its harmonics are labeled. Panel B: A heat map showing the wavenumber-frequency spectrum for the case St64. The axes and color scale are the same as in Panel A. The same labels for ridges and frequencies are present. Panel C: A heat map showing the wavenumber-frequency spectrum for the solid case (Esolid). The axes and color scale are consistent with previous panels. The labels for the convective ridge, acoustic ridge, surface wave, and resonance frequency and its harmonics are included. Panel D: A heat map showing the wavenumber-frequency spectrum for the case ESt64. The axes and color scale remain the same. The labels for the convective ridge, acoustic ridge, surface wave, and resonance frequency and its harmonics are present.
To elucidate the behaviour of the Kutta condition near the TE more clearly, figure 15 shows the distribution of the wall-pressure jump
$\varPhi _{\overset {\frown }p \overset {\frown }p}(k_x, k_z=0, \omega )$
in wavenumber–frequency space. The wall jump spectrum is obtained by performing a Fourier transform on the wall-pressure jump in both temporal and spatial directions, with a Hanning window applied in the streamwise direction and in time to prevent spectral leakage (Gloerfelt & Berland Reference Gloerfelt and Berland2013) (see (2.25)). The variables are non-dimensionalised by the displacement thickness
$\delta ^*$
and the edge velocity
$U_e$
at the centre location for the controlled region. For the solid TE case in figure 15(a), the turbulent energy is predominantly concentrated in the vicinity of the convective ridge. The acoustic content is bounded by the two lines
$U_e \pm c$
, and upstream-propagating acoustic energy evidently forms the larger proportion. In the previous study by Shubham et al. (Reference Shen, Wang, Sun, Zhang and Sun2022), the upstream acoustic spectrum under low subsonic conditions was also more energetic and less affected by Mach number changes than the downstream spectrum. For case St64 in figure 15(b), the upstream acoustic energy is largely reduced, whilst a new surface wave, propagating along the surface and decaying rapidly in the normal direction (Rienstra & Hirschberg Reference Rienstra2004; Hu et al. Reference Hu, Wan, Ye, Sun and Lu2025a
), occurs due to the impedance boundary condition. The special surface wave for airfoil TEs is first reported here, and the theoretical prediction for the wavenumber–frequency relationships at
$R=0.1$
(see (2.22)) is observed to collapse well with the numerical results. A similar distribution can be found in figures 15(c) and 15(d), with a slightly reduced convection velocity.
The integrated energy spectrum with supersonic phase velocity for solid and impedance cases. 
Furthermore, the supersonic spectrum content is integrated to facilitate a quantitative comparison, namely
\begin{equation} \phi _{sup}(\omega ) = \int _{ (\omega - U_e k_x)^2 - (k_x^2 + k_z^2)c_\infty ^2\gt 0} \varPhi _{\overset {\frown }p \overset {\frown }p}(k_x, k_z, \omega ) \,\text{d}k_x \,\text{d}k_z, \end{equation}
and the result is shown in figure 16. It is obvious that for
$\omega \delta ^*/U_e \lesssim 0.7$
, the differences between the different cases are negligible. Within the range of
$0.7 \lesssim \omega \delta ^*/U_e \lesssim 1.7$
, the impedance cases St64 and ESt64 exhibit higher spectral energy, which is attributed to the particular surface waves induced by the impedance boundary. Furthermore, in the higher-frequency range, the acoustic energy of the impedance case is significantly reduced, indicating an effective suppression of high-frequency content by the impedance wall. Notably, the difference between the extended case Esolid and the baseline case M025S is not pronounced, further suggesting that considering only the wall wavenumber–frequency spectrum distribution is insufficient to comprehensively interpret the physical mechanism for noise reduction.
Far-field noise spectrum
$\phi _{pp}$
contributed by wall-pressure jump and velocity penetration on strips of the airfoil surface: (a) separate contribution by each strip in the low-frequency range
$1\lt \textit{St}\lesssim 3$
, (b) accumulated contribution by strips from the TE
$x_{t.e.}$
to the upstream position
$x$
in the low-frequency range, (c,d) show the separate and accumulated contribution in the mid-frequency range
$3\lt \textit{St}\lesssim 5$
and (e, f) show the contribution in high-frequency range
$5\lt \textit{St}\lesssim 10$
, respectively. The light-black and light-red regions denote the areas covering the final 10 % chord length of the airfoil. The reference pressure is
$2\times 10^{-5}$
Pa. 
While the preceding content examined in detail the key factors influencing far-field radiated noise, as dictated by Amiet’s theory (Amiet Reference Amiet1976), we must also acknowledge the limitations of this consideration. These constraints are twofold. Firstly, there exist significant differences between the gust response functions of an impedance TE and a solid TE; critically, the normal Reynolds stress on the impedance wall introduces an additional dipole noise source. Secondly, the finite thickness of the airfoil TE results in a time delay (or time-retardation effect) in the acoustic waves scattered by the upper and lower surface sources. Thus, a more comprehensive noise assessment for airfoil TEs must explicitly incorporate these time-retardation effects due to varying surface source locations and account for the additional acoustic contribution arising from the wall-penetration velocity.
To address this, figure 17 presents a direct comparison of acoustic scattering efficiency using the Ffowcs Williams and Hawkings solver, with a permeable integration surface positioned on the airfoil from the TE at
$x_{t.e.}$
to the upstream location of
$x$
(Avallone et al. Reference Avallone, Van Der Velden, Ragni and Casalino2018). Specifically, the airfoil surface was divided into many streamwise strips, consistent the mesh resolution on the airfoil surface. As depicted in figure 17(a), the isolated contribution to the far-field noise from each strip is illustrated for the low-frequency range. The contribution attributed to a single strip is determined by a spanwise integral of the mesh face cells at a fixed streamwise location, averaged over the corresponding frequency range, and incorporating contributions from both the suction and pressure sides. Approaching the TE, the acoustic intensity on the airfoil surface is observed to initially increase before diminishing rapidly (Ayton et al. Reference Ayton, Gill and Peake2016). This behaviour is fundamentally due to the Kutta condition, which constrains the growth of the wall-pressure jump near the TE, thereby avoiding the singularity that would otherwise occur at that location. For case St64, the introduction of the impedance boundary condition results in a significant increase in the acoustic level, consistent with the low-frequency noise observed in figure 6. The acoustic intensity for the extended flat plate is significantly lower than for the baseline case as
$x/L \rightarrow 1$
, but higher near the TE. It is noteworthy that the extended impedance plate does not markedly alter the noise intensity compared with the case Esolid, with a slight reduction observed near the TE. To account for the interference characteristics of the sound waves radiated from the airfoil, the cumulative contribution of the airfoil strips is presented in figure 17(b). As the observation point moves upstream from the TE of the airfoil, the cumulative contribution initially increases rapidly and then grows more gradually over the airfoil surface. This behaviour again suggests that the primary noise sources are concentrated within approximately aft 10 % of the chord length near the TE (Hu et al. Reference Hu, Wan, Sun and Lu2024), and the ESt64 configuration exhibits the most significant advantage in noise reduction. Figure 17(c) presents the acoustic intensity distribution within the mid-frequency range. The results indicate that the differences between the simple solid-wall and pure impedance configurations are relatively minor. However, the acoustic power over the airfoil section of the extended impedance plate is significantly lower than that of the extended flat plate. This reduction is essentially reflected in its diminished cumulative contribution, as demonstrated in figure 17(d). For high-frequency noise in figure 17(e), the differences between the impedance and the corresponding solid cases are generally small. The extended plate’s (ETE/EITE) cumulative contribution, as shown in figure 17(f), is substantially lower than that of the conventional non-extended plate, with case ESt64’s contribution being marginally higher than that of case Esolid. To further elucidate the interference mechanisms underlying the observed noise reduction, we evaluate the destructive interference characteristics among the various source strips along both the lower and upper surface within the region
$0.7\leqslant x/L \leqslant x_{t.e.}$
. This is quantified by the cross-spectral density between the pressure contribution from an individual strip,
$p^\prime _i$
(including contributions from both the suction and pressure sides), and the integrated contribution from the entire surface,
$p^\prime _0$
\begin{equation} C_{i,0}(\omega ) = \int _{-\infty }^\infty \left [ \int _{-\infty }^\infty p^\prime _i(t) p^\prime _0(t+\tau ) \text{d}\tau \right ] e^{-i\omega t} \,\text{d}t. \end{equation}
The complex phase of
$C_{i,0}$
represents the coherence phase of each source region relative to the global acoustic radiation. Figure 18 displays the frequency-averaged cosine of this phase,
$\overline {\cos (\phi )}$
, where a value approaching
$-1$
indicates out-of-phase behaviour conducive to destructive interference. As shown in the figure 18(a), both the ETE and EITE configurations exhibit a significant reduction in
$\overline {\cos (\phi )}$
compared with the baseline and ITE cases, particularly in the aft region of the airfoil surface (
$0.8 \leqslant x \leqslant 1$
). In the moderate-to-high-frequency ranges, as shown in figures 18(b) and 18(c), the EITE case consistently reaches lower or negative values, confirming that the extended impedance design significantly promotes phase cancellation. This enhanced destructive interference effectively weakens the integrated acoustic scattering efficiency, providing a rigorous physical explanation for the superior far-field noise mitigation achieved by the extended configurations.
The frequency-averaged coherence phase of each strip on the airfoil surface: (a)
$1\lt \textit{St}\lesssim 3$
, (b)
$3\lt \textit{St}\lesssim 5$
and (c)
$5\lt \textit{St}\lesssim 10$
. 
Figure 18. Long description
The image contains three line graphs labeled (a), (b), and (c), each showing different turbulence statistics in the boundary layer. Panel A: A line graph displays the averaged wall-tangential velocity profile against the streamwise location x. The y-axis represents cos(φ), ranging from -1.0 to 1.0. Multiple lines in different colors and styles represent different cases, showing minimal discrepancies at certain streamwise locations and a pronounced downward shift in the velocity profile for the impedance case at x = 1.0. Panel B: Another line graph shows the distribution of the wall-tangential velocity fluctuation against the streamwise location x. The y-axis represents |cos(φ)|, ranging from -1.0 to 1.0. Different lines indicate various cases, with a notable bimodal profile for the extended impedance case (ESt64) at x = 1.0. Panel C: The final line graph presents the fluctuation intensity in the near-wall region against the streamwise location x. The y-axis represents |cos(φ)|, ranging from -1.0 to 1.0. Different lines show various cases, with case St64 exhibiting significantly elevated fluctuation intensity in the near-wall region.
Our prior analysis, guided by Amiet’s theory (Amiet Reference Amiet1976), reveals a contradictory initial picture: increases in the spanwise correlation length and the hydrodynamic energy of the wall-pressure jump oppose the expected noise reduction, even as decreases in the convection velocity and the supersonic phase velocity energy support an attenuating trend. To reconcile this, a comprehensive, quantitative assessment using acoustic analogy theory was performed here. Initially, by incorporating wall-penetration effects and time retardation from the finite airfoil thickness, we find an increased far-field acoustic contribution from individual surface regions, seemingly intensifying the contradiction. However, once we fully account for the interference effects among sources distributed along the airfoil TE, the resulting surface source diagnosis finally aligns with the observed far-field noise reduction trend. Fundamentally, it is recognised that noise scattering from an ETE/EITE results from multiple factors acting in concert, with this destructive interference effect providing the key insight into the noise reduction mechanism.
3.5. Vortex shedding associated with wall-pressure jump
The pressure jump across the TE is closely associated with the enforcement of the Kutta condition (Crighton Reference Crighton1985). It achieves this by both directly altering the statistical characteristics of the wall-pressure spectrum and indirectly influencing the radiated noise through its effect on vortex shedding in the vicinity of the TE. Specifically, as the wall turbulence passes through the TE, the instantaneous pressure difference initiates the vortex shedding at a rate such that the pressure jump at the TE is zero. Additionally, vortex shedding is directly related to the quadruple source in the near field. Although the scattering efficiency of quadrupole sources is usually lower than that of dipole sources, a comprehensive evaluation of the sound sources will facilitate a more systematic investigation into the noise reduction mechanisms of impedance TEs. Here, we examine three distinct source terms that are related to sound generation, namely the vortex stretching term
$S_1$
, the vortex-sound term
$S_2$
(Howe Reference Howe2003) and the longitudinal process source term
$S_3$
(Mao et al. Reference Mao, Kang, Wu, Yu, Gao, Su and Lu2020), as introduced in § 2.4.
Iso-contours of vortical source contribution near the TE for solid and impedance cases: (a) M025S, (b) St64, (c) Esolid and (d) ESt64. For each subfigure, the source terms are arranged from left to right as follows: the vortex stretching term
$\sqrt {\langle S_1^2\rangle } = \sqrt {\langle (\omega _i \omega _j \partial u_i/\partial x_j)^2\rangle }$
, the vortex-sound term
$\sqrt {\langle S_2^2\rangle }=\sqrt {\langle (\partial (\epsilon _{\textit{ijk}}\omega _i u_j)/{\partial x_k})^2\rangle }$
and the longitudinal process source term
$\sqrt {\langle S_3^2\rangle }=\sqrt {\langle {(({\partial u_i}/{\partial x_j})({\partial u_j}/{\partial x_i})-({\partial u_i}/{\partial x_i})( {\partial u_i}/{\partial x_i}))}^2\rangle }$
. The orange box highlights the distinct source distribution patterns observed in the ETE case (similarly applicable to the EITE case).
The spatial distributions of the different source terms are shown in figure 19. For the baseline case M025S (figure 19
a), the vortex-related sources are highly localised. All three source terms are predominantly concentrated immediately adjacent to the solid wall and extend along the boundary layer until the TE. The distribution pattern in the St64 case is remarkably similar to that of M025S, suggesting that the application of impedance at the solid TE does not fundamentally alter the gross spatial location or the relative strength of the primary sources. In the extended wall cases, Esolid (figure 19
c) and ESt64 (figure 19
d), a significant difference in the source distribution emerges. The distribution of the vortex stretching
$S_1$
and vortex-sound
$S_2$
terms now covers a visibly broader area, extending along the solid or porous extension. This indicates that the acoustic activity is not confined to the end of the airfoil solid section but is dispersed along the modified surface. Most notably, the longitudinal process source term
$S_3$
exhibits a distinct streamwise characteristic, as highlighted by the orange-boxed region. Specifically, the
$\sqrt {\langle S_3\rangle ^2}$
values in the vicinity of the extended section decrease markedly compared with those over the upstream airfoil surface. In contrast, the
$\sqrt {\langle S_1 \rangle ^2}$
and
$\sqrt {\langle S_2\rangle ^2}$
remain at comparable levels in both regions. In fact, the effective TE thickness, or wedge angle (Spiropoulos et al. Reference Skote and Henningson2025), is geometrically reduced by introducing the extension. This reduction effectively weakens the local vortex shedding process that is typically induced and amplified by the sharp wall-pressure jump near the TE (Ayton et al. Reference Ayton, Gill and Peake2016). This unique, attenuated spatial signature for
$S_3$
serves as compelling evidence that the longitudinal process source term is effective at highlighting the contribution of vortices directly affected by the streamwise impedance discontinuity and its associated wall-pressure jump near the airfoil TE.
Distribution of the source term in the near field: (a) root square of the vortex stretching term,
$\sqrt {\langle S_1^2 \rangle }$
, at
$x/L\simeq 0.92$
for cases M025S and St64, and
$x/L\simeq 1.04$
for the extended case, (b) the distribution of the integrated vortex stretching term along the streamwise direction, (c) root square of the local vortex-sound term,
$\sqrt {\langle S_2^2 \rangle }$
, across the boundary layer, (d) the integrated vortex-sound term, (e) root square of the longitudinal process source term,
$\sqrt {\langle S_3^2 \rangle }$
, and (f) the integrated longitudinal process source term for different cases. The light-black and light-red shaded regions denote the area covering the final 10 % chord length of the airfoil, across which the intensity of the vortex-related source term increases rapidly when viewed from downstream. 
Figure 20. Long description
The image contains six panels of line graphs depicting the distribution of source terms in the near field for different cases. Panel A: A line graph showing the root square of the vortex stretching term, S1, at various y+ values for cases M025S, St64, and the extended case. The x-axis is labeled y+ and the y-axis is labeled S1². Panel B: A line graph showing the distribution of the integrated vortex stretching term along the streamwise direction, x/L. The x-axis is labeled x/L and the y-axis is labeled ∫S1²dy. Panel C: A line graph showing the root square of the local vortex-sound term, S2, across the boundary layer for different cases. The x-axis is labeled y+ and the y-axis is labeled S2². Panel D: A line graph showing the integrated vortex-sound term along the streamwise direction, x/L. The x-axis is labeled x/L and the y-axis is labeled ∫S2²dy. Panel E: A line graph showing the root square of the longitudinal process source term, S3, at various y+ values for different cases. The x-axis is labeled y+ and the y-axis is labeled S3². Panel F: A line graph showing the integrated longitudinal process source term along the streamwise direction, x/L. The x-axis is labeled x/L and the y-axis is labeled ∫S3²dy. The light-black and light-red shaded regions denote the area covering the final 10 percent chord length of the airfoil, across which the intensity of the vortex-related source term increases rapidly when viewed from downstream. Labels: M025S; St64; Esolid; ESt64.
A quantitative comparison of different source terms within the boundary layer is further shown in figure 20. Figure 20(a) illustrates the variation of the vortex stretching term at different heights within the boundary layer. The sampling location is positioned slightly upstream of the TE (
$x/L=x_{t.e.}-0.08$
) to obtain a relatively smooth source term distribution: specifically at
$x/L = 0.92$
for the M025S and St64 cases, and at
$x/L = 1.04$
for the extended cases (ETE/EITE). The results show that all cases peak sharply at
$y^+ \simeq 3$
and decay rapidly at higher positions. Additionally, the impedance wall and the extended plate exhibit entirely opposite effects: the impedance wall causes a moderate increase in the vortex stretching term, whereas the extended plate suppresses vortex deformation. When these two mechanisms are combined in case ESt64, the suppressive effect of the extended plate plays a dominant role. Figure 20(b) displays the integrated value of the vortex stretching term within the boundary layer. As the flow develops downstream, the vortex stretching term gradually weakens, and the extended plate further accentuates this trend. In other words, configurations with an extended solid or impedance plate tend to exhibit lower intensity near the TE. Figure 20(c) compares the distribution characteristics of the vortex-sound source term, which exhibits two distinct peaks within the boundary layer: one near the wall and the other at
$y^+ \simeq 8$
for all cases. The vortex-sound source term also shows a reduction in intensity across most regions, except for the immediate near-wall region (
$y^+ \lesssim 3$
). The integrated results shown in figure 20(d) reveal a trend highly consistent with that observed in figure 20(b). Moreover, figure 20(e) demonstrates the distribution of the recently developed longitudinal process source term within the boundary layer (Mao et al. Reference Mao, Kang, Wu, Yu, Gao, Su and Lu2020). The introduction of the extended impedance wall results in an enhancement of the source term compared with that of the ETE, although its intensity remains significantly lower than that of the baseline case. Concurrently, the source term’s peak position shifts outward to approximately
$y^+ \simeq 30$
, nearly coinciding with the peak location of the Reynolds stress (see figure 10). Beyond this peak, the impedance boundary has almost no noticeable effect, indicating that flow penetration at the wall surface primarily influences the immediate near-wall regions (Scalo et al. Reference Sano, Abreu, Cavalieri and Wolf2015; Chen & Scalo Reference Chen and Scalo2021b
). Furthermore, in the streamwise direction (figure 20
f), the source term gradually increases, peaks at the TE (
$x\simeq L$
), and then decays rapidly downstream of this position. Notably, the longitudinal process source term,
$S_3$
, shows again significant potential for emphasising the suppression of the vortex shedding process. Thus, a consistent trend emerges despite differences in source term definitions: extending the plate significantly reduces the overall source term intensity at the TE compared with the baseline case. While the EITE causes a slight local increase in vortex-related source terms directly near the wall surface, this effect is ultimately subordinate to the overall suppression, which benefits far-field noise reduction.
4. Summary and conclusion
This study performed high-fidelity implicit large eddy simulation to investigate the flow and acoustic radiation of a NACA-0012 airfoil at a low Mach number and a moderate Reynolds number. Drawing inspiration from the Kutta condition, we developed an extended TE configuration (ETE/EITE) capable of significantly reducing the pressure jump across the upper and lower surfaces near the edge. Furthermore, the influence of the impedance boundary condition on the wall-pressure jump spectrum and the resultant vortex shedding process at the TE was detailed from the perspective of the Kutta condition. The comprehensive investigation elucidates the physical mechanism underlying the noise reduction achieved by the ETE/EITE, and a more detailed summary is given as follows.
Our findings reveal that, while the simple ITE increases far-field noise as its characteristic frequency approaches the most unstable perturbation frequency, this enhancement progressively weakens with higher characteristic frequencies. In contrast, the design of ETE demonstrates a notable noise reduction of 3.62 dB, a benefit that the extended impedance design (EITE) further amplifies to 4.22 dB. This noise reduction is observed across the entire studied frequency range. Furthermore, analysis of turbulence statistics reveals that the EITE induces a wall-normal flow penetration near the TE, leading to a decrease in the intensity of both the streamwise and spanwise Reynolds stresses, as well as a significant reduction in near-wall pressure fluctuations. Subsequent SPOD analysis also confirms that the coherent structures decay more rapidly in the presence of the extended impedance wall.
The unsteady Kutta condition, which governs TE noise, is directly related to the wall-pressure jump and vortex shedding. The ETE is found to enhance the destructive interference between sound waves scattered from sources near the TE, which is consistent with the observed noise reduction. Further analysis of vortex-related source terms reveals that the extended plate effectively suppresses vortex shedding. For the EITE, we present the first detailed wavenumber–frequency characterisation of the convective, acoustic and surface wave components in its wall-pressure jump spectrum, which is also confirmed by our newly improved theoretical model. The findings indicate that the EITE achieves even greater noise reduction by further decreasing the convection velocity of pressure fluctuations and weakening supersonic components, thereby amplifying this destructive interference effect.
In conclusion, the Kutta condition at the TE serves as an effective principle for guiding low-noise TE design while offering meaningful insights into the associated noise mitigation mechanisms. Specifically, extending the TE with either a solid or impedance plate effectively modifies the local wall-pressure jump and suppresses vortex shedding, thereby significantly reducing the acoustic scattering efficiency of the TE. The extended impedance configuration achieves the most significant noise reduction, albeit at a marginal cost to aerodynamic performance. Specifically, while the absolute drag force increases by approximately 5.89 %, the drag coefficient actually decreases by 5.45 % due to the expansion of the wetted surface area, as shown in Appendix D. These results further emphasise the inherent challenge in simultaneously optimising aeroacoustic and aerodynamic performance, necessitating more comprehensive multi-objective optimisation in future designs. It is important to note that the impedance model used in this study represents a simplified acoustic liner instead of a general porous medium. Future research should explore the use of an extended finite-thickness porous plate to further allow for effective flow interaction between the upper and lower surfaces (Ali et al. Reference Ali, Azarpeyvand and Da Silva2018; Teruna et al. Reference Teruna, Avallone, Ragni and Casalino2021b ), which may lead to better noise reduction performance.
Acknowledgements
The numerical computations were performed at Hefei Advanced Computing Centre.
Funding
This work was supported by the National Natural Science Foundation of China under Grant Nos. 12322210, 12402317 and 12388101. The first author also acknowledges financial support from the Fundamental Research Funds for the Central Universities (Grant No. WK2090000073) and the Science Foundation for Young Scientists of State Key Laboratory of High Temperature Gas Dynamics (Grant No. 2025QN06).
Declaration of interests
The authors report no conflicts of interest.
Appendix A. Convergence of numerical simulations
For simulating airfoil flow at a moderate Reynolds number, meeting the convergence requirements for both the spanwise domain size and the statistical averaging duration is crucial, despite the high computational cost. Figure 21(a) illustrates the spanwise correlation of pressure fluctuations near the airfoil TE. For all cases, the correlation exhibits an approximately exponential decay, with the coefficient dropping to 0.01 at a spanwise separation of only approximately
$0.02L$
(Wolf et al. Reference Wang, Hao, Lu, Tong and Song2012). This confirms that the current spanwise domain width is sufficient to ensure the flow is fully resolved. Furthermore, figure 21(b) demonstrates the statistical convergence of pressure sampled at a near-field location
$R_s=4L$
above the TE in the middle
$x{-}y$
plane. The flow statistics achieve excellent convergence within approximately 10 flow-through times. While we omit cases with lower impedance characteristic frequencies, which typically feature large-scale vortex shedding and thus demand longer statistical times for low-frequency convergence, the present duration remains fundamentally adequate for these requirements as well.
Convergence validation of computational parameters: (a) the spanwise correlation of wall-pressure fluctuation. The measurement location is
$x/L=0.95$
for the baseline (M025S) and simple impedance (St64) cases, and
$x/L=1.07$
for the extended cases; (b) convergence history of the time-averaged pressure sampled at the near-field location
$(1, 4)$
in the central
$x{-}y$
plane, showing the 95 % confidence interval. 
Appendix B. Implementation of the time-domain impedance boundary conditions
Accurate TDIBCs are essential for simulating turbulence flow over acoustic liners (Fung & Ju Reference Fung and Ju2004; Scalo et al. Reference Sano, Abreu, Cavalieri and Wolf2015; Douasbin et al. Reference Douasbin, Scalo, Selle and Poinsot2018). The fundamental relationship in the frequency domain is defined as
\begin{equation} \hat {p} = Z(\omega ) \hat {v_n}. \end{equation}
To accommodate the frequency-dependent boundary condition within the time-stepping framework of the Navier–Stokes equations, the following algorithm is utilised.
First, we define the up- and down-travelling waves as
\begin{equation} \left \{\! \begin{aligned} v^+ &= v_n^{\prime } + p^{\prime },\\ v^- &= v_n^{\prime } - p^{\prime },\\ \end{aligned} \right . \end{equation}
where
$v_n^{\prime }$
and
$p^{\prime }$
are the fluctuating wall-normal velocity and pressure, respectively.
During the computation, the up-travelling wave at the next time step,
$v^+(t+\Delta t)$
, is evaluated using a first-order Taylor expansion as (where
$y$
denotes the local wall-normal direction)
\begin{equation} \begin{aligned} v^+(t+\Delta t) &= p^{\prime }(y-c_\infty \Delta t, t) + v_n^{\prime }(y-c_\infty \Delta t, t) \\ &\simeq \left [p^{\prime }(y, t) + v_n^{\prime }(y, t)\right ] - c_\infty \Delta t \frac {\partial }{\partial y} \left [p^{\prime }(y, t) + v_n^{\prime }(y, t)\right ]\!. \end{aligned} \end{equation}
The down-travelling signal
$\hat {v}^-(\omega )$
is deduced in the frequency domain, given by
\begin{equation} \hat {v}^-(\omega ) = -\hat {v}^+(\omega ) + \hat {\tilde {W}}(\omega ) \hat {v}^+(\omega ), \end{equation}
where
$\hat {\tilde {W}}(\omega )=2/(1+Z(\omega ))$
is the reflection coefficient.
Transforming this into the time domain yields the convolution integral
\begin{equation} v^-(t) = -v^+(t) + \int _0^{\infty } \tilde {W}(\tau ) v^+(t-\tau ) \text{d}\tau . \end{equation}
By applying the residue method to evaluate the integrand at time
$t+\Delta t$
, the down-travelling wave can be decomposed into a summation of discrete contributions from each pole such that
\begin{equation} v^-(t+\Delta t) = -v^+(t+\Delta t) + \sum _k v_k^-(t+\Delta t), \end{equation}
where each component
$v_k^-(t+\Delta t)$
is calculated recursively to avoid storing the entire time history. The integration process is summarised as
\begin{equation} \begin{aligned} v_k^-(t+\Delta t) &= \int _0^{\infty } \mu _k e^{i\lambda _k \tau } v^+(t+\Delta t-\tau ) \text{d}\tau \\ &= \int _{\Delta t}^{\infty }\mu _k e^{i\lambda _k (u+\Delta t)}v^+(t-u)\text{d}u\\ &= \int _{0}^{\infty }\mu _k e^{i\lambda _k (u+\Delta t)}v^+(t-u)\text{d}u + \int _{-\Delta t}^{0}\mu _k e^{i\lambda _k (u+\Delta t)}v^+(t-u)\text{d}u\\ &= e^{i\lambda _k \Delta t} v_k^-(t) + \int _{0}^{\Delta t} \mu _k e^{i\lambda _k \tau } v^+(t+\Delta t - \tau ) \text{d}\tau \\ &\simeq z_k v_k^-(t) + \mu _k \Delta t \left [w_{k0} v^+(t+\Delta t) + w_{k1} v^+(t)\right ], \end{aligned} \end{equation}
where
$\lambda _k$
represents the
$k$
th pole and
$\mu _k$
is the corresponding residue
\begin{equation} \mu _k = i \textit{Res} \left [ \frac {2}{1+Z(\omega )}, \lambda _k \right ]\!. \end{equation}
The numerical weights
$w_{k0}$
and
$w_{k1}$
are defined as
\begin{equation} \left \{ \begin{aligned} w_{k0} &= -\frac {z_k-1}{\lambda _k^2 \Delta t^2} - \frac {1}{i\lambda _k \Delta t}, \\ w_{k1} &= \frac {z_k-1}{\lambda _k^2 \Delta t^2} + \frac {z_k}{i\lambda _k \Delta t}. \end{aligned} \right . \end{equation}
Finally, the physical primitive variables at the new time step are reconstructed from the characteristic waves via
\begin{equation} \left \{ \begin{aligned} v_n^{\prime }(t+\Delta t) &= \frac {1}{2} \left [ v^+(t+\Delta t) + v^-(t+\Delta t) \right ],\\ p^{\prime }(t+\Delta t) &= \frac {1}{2} \left [ v^+(t+\Delta t) - v^-(t+\Delta t) \right ]\!. \end{aligned} \right . \end{equation}
The implementation of the impedance boundary condition is validated using a one-dimensional linear wave propagating within an impedance tube (Tam & Auriault Reference Sun, Wang, Du and Sun1996; Scalo et al. Reference Sano, Abreu, Cavalieri and Wolf2015). The specific impedance parameters are given by
\begin{equation} R = 0.2, \quad \omega _r = 40, \quad \zeta = 0.6. \end{equation}
The computational domain, defined as
$-1 \leqslant x \leqslant 1$
, is discretised into 2000 elements to ensure sufficient numerical resolution. An initial disturbance is prescribed as
\begin{equation} p^\prime (x,0)=\frac {1}{2}A e^{-\chi k_x^2 x^2} \cos (2\pi k_x x), \quad v_n^\prime (x,0)=p^\prime (x,0), \end{equation}
where the amplitude and shape parameters are
$A=10^{-6}$
,
$\chi =0.8$
and
$k_x=7$
. Regarding the boundary treatment, the impedance condition is applied at the right boundary, while a non-reflective outgoing condition is maintained at the left boundary.
The numerical results at
$t=2.0$
are presented in figure 22. The excellent agreement between the current simulation and the reference data demonstrates the accuracy and robustness of the present TDIBC algorithm. Furthermore, the performance of this impedance condition has been successfully validated in a three-dimensional channel flow at a moderate friction Reynolds number (Hu et al. Reference Hu, Wan, Ye, Sun and Lu2025a
,
Reference Hu, Wan, Zhang, Sun and Lub
), confirming its applicability to complex turbulent flow.
Planar up-travelling wave with impedance boundary contition at
$t=2.0$
.
Appendix C. Effect of wetted surface variation on far-field noise
To ensure a rigorous comparison between the baseline and extended configurations, the influence of the modified effective chord length and the associated chord-based Reynolds number must be considered. The addition of the TE extension increases the effective chord by 12 %, which introduces a marginal global scaling shift. According to the far-field scaling for TE noise (see (2.30)), Reynolds number effects influence the radiated sound primarily through the characteristic boundary-layer thickness, rather than directly through the chord length. For a turbulent boundary layer, the empirical relationship,
$\delta /L \propto Re^{-1/5}$
(Eckert & Drake Reference Eckert and Drake1959; Amiet Reference Amiet1976), yields
$\delta \propto L^{4/5}$
. Therefore, increasing the effective chord length by
$12\,\%$
produces only a minor variation in
$\delta$
, and the associated change in sound pressure level can be estimated as
$\Delta \textit{SPL}_{1/3}=10\log _{10}(1.12^{4/5}) \simeq 0.39\,\mathrm{dB}$
. This minor increase is significantly outweighed by the observed noise reduction of 4.22 dB, suggesting that the acoustic benefit of the impedance treatment would be even more pronounced if evaluated against a baseline of identical chord length. Furthermore, since noise suppression mechanisms are dominated by local interference effects and the change of the surface pressure jump at the edge (Teruna et al. Reference Teruna, Avallone, Ragni, Rubio-Carpio and Casalino2020, Reference Teruna, Avallone, Ragni and Casalino2021b
), these results remain representative of the fundamental physics across moderate variations in Reynolds number.
Appendix D. Effects on the aerodynamic performance
Since the present study considers a NACA-0012 airfoil at a zero-degree angle of attack, the lift produced is inherently zero. Consequently, our analysis focuses on the impact of the TE modifications on the total drag,
$F$
. The drag coefficient,
$C_D$
, is defined as
\begin{equation} C_{\!D}= \frac {F}{\frac {1}{2}\rho _\infty U_\infty ^2 L_x L_z}= \frac {\displaystyle\iint \!(\tau _w \cos \theta + p \sin \theta )\,\text{d}s \,\text{d}z}{\frac {1}{2}\rho _\infty U_\infty ^2 L_x L_z}= \frac {\displaystyle\iint\! (C_{\!f} \cos \theta + C_{\!p} \sin \theta )\,\text{d}s\,\text{d}z}{L_x L_z}, \end{equation}
where
$\theta$
is the angle between the airfoil tangential direction and the
$x$
-axis,
$s$
is the coordinate along the airfoil surface and
$L_x$
represents the total streamwise extent of the airfoil configuration. As shown in table 2, the St64 (ITE) case maintains nearly the same
$C_D$
as the baseline M025S case. For the extended TE cases (Esolid and ESt64), the total drag force increases by approximately 6 %. However, the drag coefficient actually decreases by nearly 6 % due to the increased reference length
$L_x$
. This observation is consistent with figure 9, where the extended designs significantly alter the local distributions of the pressure and skin-friction coefficients.
Drag and drag coefficient for the airfoil with solid or impedance TEs.
x/L∈[0.15,0.20]
z/L∈[0.01,0.11]
Lp
Δt
ts
R
ζ
fr
a∼c
Cp
Cf
x/L≃0.85
x/L≃0.85
x/L≃1.05
x/L≃1.05
x/L≃0.91
Rs=8L
Rs=4L
1
3
5
ut
vn
p
Cp
Cf
δ∗
θ
Cη=ρ¯v′′v′′~/Q
(pss−pps)rms/Q
pss
pps
x/L
u~t+
(ut′′)rms+
(vn′′)rms+
−(ut′′vn′′~)+
(w′′)rms+
(p′)rms+
Qe=1/2ρeUe2
L/Ue
τw
L/uτ
τw
δ∗/Ue
τw
ν/uτ
R=0.1
ϕpp
1
xt.e.
x
3
5
2×10−5
1
3
5
⟨S12⟩=⟨(ωiωj∂ui/∂xj)2⟩
⟨S22⟩=⟨(∂(ϵijkωiuj)/∂xk)2⟩
⟨S32⟩=⟨((∂ui/∂xj)(∂uj/∂xi)−(∂ui/∂xi)(∂ui/∂xi))2⟩
⟨S12⟩
x/L≃0.92
x/L≃1.04
⟨S22⟩
⟨S32⟩
x/L=0.95
x/L=1.07
(1,4)
x−y
t=2.0
