1. Introduction
In recent years the rapid development of commercial aviation has elevated flow-induced noise to a prominent environmental concern, with considerable implications for the residential quality of life. The sources of aviation noise are manifold, among which turbulent boundary layer trailing-edge (TBL-TE) noise constitutes an important component of aerofoil self-noise. The TBL-TE noise exhibits distinct broadband characteristics, leading researchers to investigate a range of control strategies (Lee et al. Reference Lee, Ayton, Bertagnolio, Moreau, Chong and Joseph2021; Doolan & Moreau Reference Doolan and Moreau2022). These noise control methods can be categorised into two groups based on the necessity of external energy input: passive control and active control. Passive control has attracted considerable academic interest due to its operational simplicity and cost effectiveness. Typical passive approaches include trailing-edge serrations (Jones & Sandberg Reference Jones and Sandberg2012; Chong & Vathylakis Reference Chong and Vathylakis2015; Lyu, Azarpeyvand & Sinayoko Reference Lyu, Azarpeyvand and Sinayoko2016; Huang Reference Huang2017; Avallone et al. Reference Avallone, Van Der Velden, Ragni and Casalino2018; Ayton Reference Ayton2018; Pereira et al. Reference Pereira, Avallone, Ragni and Scarano2022), porous media (Jaworski & Peake Reference Jaworski and Peake2013; Kisil & Ayton Reference Kisil and Ayton2018; Carpio et al. Reference Carpio, Martínez, Avallone, Ragni, Snellen and van der Zwaag2019; Teruna et al. Reference Teruna, Manegar, Avallone, Ragni, Casalino and Carolus2020), finlets (Clark et al. Reference Clark, Alexander, Devenport, Glegg, Jaworski, Daly and Peake2017; Gstrein, Zang & Azarpeyvand Reference Gstrein, Zang and Azarpeyvand2023; Pan & Ye Reference Pan and Ye2025) and biomimetic coatings (Zhou, Zhong & Zhang Reference Zhou, Zhong and Zhang2021). In contrast, active TBL-TE noise control research remains in its early stages (Szőke et al. Reference Szőke, Fiscaletti and Azarpeyvand2020a ). The advantages of the active method lie in its capacity to achieve flexible and efficient flow and noise control through minimal energy input.
Among active control approaches, blowing/suction has attracted increasing attention for its effectiveness in promoting aerodynamic performance. The early work of Park & Choi (Reference Park and Choi1999) showed that uniform blowing elevates the streamwise vortices and reduces the skin friction in the vicinity of the control slot. However, the skin friction downstream of the blowing slot increases due to enhanced vortices. Kametani & Fukagata (Reference Kametani and Fukagata2011), Kametani et al. (Reference Kametani, Fukagata, Örlü and Schlatter2015) designed a distributed active control strategy within the turbulent boundary layer. According to their findings, uniform blowing substantially reduces the skin friction drag and intensifies turbulent fluctuations, whereas uniform suction has the opposite effect. Kametani & Fukagata (Reference Kametani and Fukagata2011), based on the FIK identity (Fukagata, Iwamoto & Kasagi Reference Fukagata, Iwamoto and Kasagi2002), demonstrated that the mean convection term plays a critical role in drag reduction, as revealed through the decomposition of the skin friction coefficient. Mahfoze et al. (Reference Mahfoze, Moody, Wynn, Whalley and Laizet2019) further optimised the intensity and spatial extent of the blowing. Additionally, some studies (Antonia, Zhu & Sokolov Reference Antonia, Zhu and Sokolov1995; Oyewola, Djenidi & Antonia Reference Oyewola, Djenidi and Antonia2003) indicated that, under moderate momentum thickness Reynolds numbers (
$Re_{\theta } \lt 660$
), sufficiently strong suction can relaminarise the boundary layer. In such cases, the skin friction coefficient is markedly reduced. Some recent studies on aerofoils also confirm the potential of active control techniques (Zha et al. Reference Zha, Ren, Gan and Espinal2019; Xu & Zha Reference Xu and Zha2020; Fahland et al. Reference Fahland, Stroh, Frohnapfel, Atzori, Vinuesa, Schlatter and Gatti2021; Kornilov, Kavun & Popkov Reference Kornilov, Kavun and Popkov2021; Quadrio et al. Reference Quadrio, Chiarini, Banchetti, Gatti, Memmolo and Pirozzoli2022).
In the past decade, substantial efforts have been made to leverage blowing/suction techniques for trailing-edge noise mitigation. Although no direct measurements of far-field noise were conducted, the results of the flat plate experiments (Wolf et al. Reference Wolf, Lutz, Würz, Krämer, Stalnov and Seifert2015; Szőke et al. Reference Szőke, Fiscaletti and Azarpeyvand2020a ) showed that suction provides significant benefits in reducing trailing-edge noise. Specifically, Wolf et al. (Reference Wolf, Lutz, Würz, Krämer, Stalnov and Seifert2015) reported that flow suction reduces velocity gradients and vertical velocity fluctuations within the boundary layer. The experiments of Szőke et al. (Reference Szőke, Fiscaletti and Azarpeyvand2020a ) further showed that the decrease in the logarithmic layer extent and turbulent energy is responsible for the decrease in mid-frequency surface pressure fluctuations, while the increase in buffer layer energy leads to an increase in high-frequency pressure spectra. Their findings also suggest that there exists an optimal suction angle. Numerical simulations by Cesar, Kamliya Jawahar & Azarpeyvand (Reference Cesar, Kamliya Jawahar and Azarpeyvand2018) demonstrated that suction applied upstream of a NACA0012 aerofoil trailing edge reduces turbulent kinetic energy in the boundary layer and wake, thereby attenuating far-field noise. Moreover, previous findings (Cesar et al. Reference Cesar, Kamliya Jawahar and Azarpeyvand2018; Szőke et al. Reference Szőke, Fiscaletti and Azarpeyvand2020a ) also demonstrated that the greater the suction intensity, the more pronounced the resulting noise attenuation. Currently, it remains challenging to achieve optimal control effects with a limited suction intensity. Blowing control has also emerged as a promising strategy for noise reduction. Szőke et al. (Reference Szőke, Fiscaletti and Azarpeyvand2018) designed a spanwise spaced, inclined transverse jet array to mitigate trailing-edge noise, and the results showed that the control effectiveness depends on blowing intensity and angle. Szőke et al. (Reference Szőke, Fiscaletti and Azarpeyvand2020b ) further investigated the noise reduction efficacy of uniform blowing applied upstream of a flat plate trailing edge. Their experimental results showed that blowing reduces the energy content within the turbulent boundary layer and pushes the turbulent structures away from the wall, ultimately resulting in a noise reduction in the medium-to-high frequency range. Investigations by Moreau et al. (Reference Moreau, Laffay, Idier and Atalla2016), Manegar et al. (Reference Manegar, Stahl, Carolus and Binois2019) also demonstrated that appropriate blowing on the aerofoil can effectively reduce low-frequency noise. Additionally, some researchers have attempted to apply blowing at the blunt trailing edge of an aerofoil to emulate the function of a splitter plate. Guo et al. (Reference Guo, Maryami, Yang, Yang, Wang and Liu2023) and Yang et al. (Reference Yang, Arcondoulis, Yang, Guo, Maryami, Bi and Liu2023) demonstrated that this control method can stabilise the shear layer and reduce the turbulent kinetic energy near the wake, thereby contributing to noise reduction. Yang et al. (Reference Yang, Liu, Zhang and Bi2024) further combined blowing with trailing-edge serrations to achieve additional noise attenuation. Arnold et al. (Reference Arnold, Lutz, Krämer and Rautmann2018) applied a combined blowing and suction control to reduce wind turbine blade noise by 3.6 dB. Kumar & Bhumkar (Reference Kumar and Bhumkar2024) demonstrated that periodic suction–blowing excitation can modify tonal noise in laminar boundary layers. Despite blowing/suction methods exhibiting potential in noise reduction, a comprehensive understanding of their noise reduction performance and underlying physical mechanisms remains incomplete, particularly regarding TBL-TE noise.
In engineering, a primary objective is to maximise control effectiveness while simultaneously minimising energy input. Over the years, numerous optimisation methods have been developed to achieve this objective. Widely used optimisation methods include adjoint optimisation (Schulze & Sesterhenn Reference Schulze and Sesterhenn2013), resolvent analysis (Liang et al. Reference Liang, Wan, She, Zhang, Sun and Lu2024), genetic algorithm (Zigunov, Sellappan & Alvi Reference Zigunov, Sellappan and Alvi2022), reinforcement learning (Li & Zhang Reference Li and Zhang2022), amongst others. Recently, a machine learning approach known as Bayesian optimisation, designed to address expensive black-box function optimisation problems, has garnered considerable attention within the fluid dynamics community (Chaitanya & Vellanki Reference Chaitanya and Vellanki2021; Han, Huang & Xu Reference Han, Huang and Xu2023; Reumschüssel et al. Reference Reumschüssel, Li, zur Nedden, Wang, Noack and Paschereit2024). Bayesian optimisation is a derivative-free method that maintains a surrogate model to predict the probability distribution of the objective function while employing an acquisition function capable of balancing exploration and exploitation for decision making (Shahriari et al. Reference Shahriari, Swersky, Wang, Adams and De Freitas2015). To date, the Bayesian optimisation algorithm has been successfully applied across numerous flow control fields, including drag reduction design (Mahfoze et al. Reference Mahfoze, Moody, Wynn, Whalley and Laizet2019; Blanchard et al. Reference Blanchard, Cornejo Maceda, Fan, Li, Zhou, Noack and Sapsis2021; Han et al. Reference Han, Huang and Xu2023), jet mixing enhancement (Blanchard et al. Reference Blanchard, Cornejo Maceda, Fan, Li, Zhou, Noack and Sapsis2021; Li et al. Reference Li, Noack, Wang, Cornejo Maceda, Pickering, Shaqarin and Tyliszczak2024; Reumschüssel et al. Reference Reumschüssel, Li, zur Nedden, Wang, Noack and Paschereit2024) and transition delay (Pershin et al. Reference Pershin, Beaume, Eaves and Tobias2022). However, the application of Bayesian optimisation within the field of aeroacoustics remains comparatively limited. Chaitanya & Vellanki (Reference Chaitanya and Vellanki2021) utilised Bayesian optimisation algorithms to optimise the blade shape for the reduction of aerofoil–turbulence interaction noise. Moreover, Lim et al. (Reference Lim, Garbo, Bekemeyer, Appel, Ewert and Delfs2022) conducted multi-objective Bayesian optimisation addressing both aerodynamics and aeroacoustics. It should be noted that, owing to computational cost constraints, the aforementioned aeroacoustic studies employed two-dimensional (2D) Reynolds-averaged Navier–Stokes simulations, which may yield less accurate far-field noise estimations. Although Bayesian optimisation algorithms are already highly efficient, the computational cost associated with conducting high-fidelity simulations – such as large eddy simulations – during each sampling stage remains substantial. This is primarily due to the stringent requirements for numerical simulation accuracy and statistical reliability in far-field noise assessment. Currently, reducing the computational burden of the optimisation process remains a significant challenge. One promising approach to addressing this issue involves controlling sound sources based on the acoustic theory, as it enables the avoidance of direct far-field noise calculations during the optimisation process.
This study aims to reduce aerofoil TBL-TE noise through the application of judicious blowing/suction. A parallel Bayesian optimisation algorithm is used to determine the optimal control positions and blowing/suction angles. For each sampling, three-dimensional (3D) large eddy simulations and an acoustic analogy are performed to compute far-field noise. Furthermore, the present study also proposes a sound-source-oriented coarse-grained Bayesian optimisation framework based on multi-process acoustic theory (Mao et al. Reference Mao, Kang, Wu, Yu, Gao, Su and Lu2020; Hu et al. Reference Hu, Wan, Ye, Sun and Lu2022; Yu et al. Reference Yu, Wan, Hu, Sun and Lu2025) to reduce computational costs associated with the optimisation process. In addition, the noise reduction mechanisms of the optimal control scenario are investigated through an analysis of modifications in flow structures and noise sources, thereby enhancing the understanding of active flow control techniques.
The remainder of this study is organised as follows. Section 2 provides an introduction to the numerical methods and acoustic theories. The geometry configuration, mesh, simulation details and validations are detailed in § 3. Section 4 presents the optimisation results and the corresponding noise reduction effect. Changes in flow structures and noise suppression mechanisms are also studied by comparing different cases. Finally, § 5 summarises the main findings and provides the conclusion.
2. Numerical methods and acoustic theories
2.1. The governing equations
The flow past a NACA0012 aerofoil with or without trailing-edge blowing or suction is solved based on the fully compressible Navier–Stokes equations, which can be non-dimensionalised by the aerofoil chord length
$l_c$
, free-stream density
$\rho _0$
and speed of sound
$c_0$
. The non-dimensional governing equations are expressed as
\begin{align} \frac {\partial \rho }{\partial t}+\frac {\partial }{\partial x_{\kern-1pt j}} (\rho u_{\kern-1pt j})=0, \\[-28pt] \nonumber \end{align}
\begin{align} \frac {\partial }{\partial t}\left (\rho u_{\kern-1pt i}\right )+\frac {\partial }{\partial x_{\kern-1pt j}} (\rho u_{\kern-1pt i} u_{\kern-1pt j})=-\frac {\partial p}{\partial x_i}+\frac {\partial }{\partial x_{\kern-1pt j}} (\tau _{\textit{ij}}), \\[-28pt] \nonumber \end{align}
\begin{align} \frac {\partial }{\partial t}(\rho E)+\frac {\partial }{\partial x_{\kern-1pt j}} (\rho u_{\kern-1pt j} E)=-\frac {\partial }{\partial x_{\kern-1pt j}} (p u_{\kern-1pt j})+\frac {\partial }{\partial x_{\kern-1pt j}} (\tau _{\textit{ij}} u_{\kern-1pt i} )-\frac {\partial q_j}{\partial x_{\kern-1pt j}}, \\[2pt] \nonumber \end{align}
where
$E=p /[\rho (\gamma -1)]+u_{\kern-1pt j} u_{\kern-1pt j} / 2$
represents the total energy,
$p$
denotes the pressure,
$\rho$
is the density and
$u_{\kern-1pt i}$
are the velocity components. The viscous stress
$\tau _{\textit{ij}}$
and the heat flux term
$q_j$
are treated as
\begin{align} \tau _{\textit{ij}}=\frac {\mu \textit{Ma}}{\textit{Re}}\left (\frac {\partial u_{\kern-1pt i}}{\partial x_{\kern-1pt j}}+\frac {\partial u_{\kern-1pt j}}{\partial x_i}-\frac {2}{3} \frac {\partial u_m}{\partial x_m} \delta _{\textit{ij}}\right )\!, \\[-28pt] \nonumber \end{align}
\begin{align} q_j=-\frac {\mu \textit{Ma}}{(\gamma -1) \textit{Pr}\textit{Re}} \frac {\partial T}{\partial x_{\kern-1pt j}}\!, \\[2pt] \nonumber \end{align}
where the Prandtl number
$Pr$
and the ratio of gas-specific heat
$\gamma$
are taken as 0.72 and 1.4, respectively. Here
$Re = \rho U_0 l_c / \mu _0$
represents the Reynolds number,
$Ma$
denotes the Mach number and
$T$
is the temperature. The non-dimensional dynamic viscosity coefficient
$\mu$
is simplified to a constant 1 at the low free-stream velocity, and the equation of state for an ideal gas
$p = \rho T / \gamma$
is used to close the equations. In the sponge zone shown in figure 1, we add an artificial damping term to the right-hand side of each governing equation to suppress spurious acoustic reflections, i.e.
\begin{align} D_q = -A(r)(\boldsymbol{q} - \boldsymbol{q}_0), \\[-28pt] \nonumber \end{align}
\begin{align} A(r)=B\left \{1+{\rm e}^{\left (\frac {1}{t-1}+\frac {1}{t}\right )}\right \}^{-1}, \\[2pt] \nonumber \end{align}
where
$t= (r-r_s ) / (r_e-r_s )$
;
$r_s$
and
$r_e$
denote the starting and ending locations of the sponge zone, respectively. The maximum damping strength
$B$
is taken as
$0.05$
. Here
$\boldsymbol{q}$
and
$\boldsymbol{q}_0$
are conservation variables and far-field reference quantities. The implicit large eddy simulations (ILES) were conducted based on the open-source solver PyFR (Witherden, Farrington & Vincent Reference Witherden, Farrington and Vincent2014), using a third-order flux reconstruction for spatial discretisation. The third-order total variation diminishing Runge–Kutta scheme is used for time advancement. Additionally, all the simulations are conducted on the central processing unit (CPU)/deep computing unit (DCU) parallel heterogeneous platform.
Sketch of the computational zone and the mesh topology: (a) the computational zone, including the sponge zone, FW–H integrated face and the far-field observers, (b) the global view of the mesh topology, (c) the enlarged view near the zig–zag strip, and (d) the enlarged view near the trailing edge of the NACA0012 aerofoil.
2.2. The Ffowcs–Williams Hawkings acoustic analogy
In this study, the far-field noise is predicted by solving the Ffowcs–Williams Hawkings (FW–H) acoustic analogy equation. The convective form of the FW–H equation is employed (Najafi-Yazdi, Brès & Mongeau Reference Najafi-Yazdi, Brès and Mongeau2011), i.e.
\begin{align} \begin{gathered} {\left [\frac {\partial ^2}{\partial t^2}-c_0^2 \frac {\partial ^2}{\partial x_i \partial x_{\kern-1pt j}}+2 U_{0 j} \frac {\partial ^2}{\partial t \partial x_{\kern-1pt j}}+U_{0 i} U_{0 j} \frac {\partial ^2}{\partial x_i \partial x_{\kern-1pt j}}\right ]\left [H(f) \rho ^{\prime }\right ]=} \\ \left (\frac {\partial }{\partial t}+U_{0 j} \frac {\partial }{\partial x_{\kern-1pt j}}\right )\left [Q_k n_k \delta (f)\right ]-\frac {\partial }{\partial x_i}\left [L_{\textit{ij}} n_j \delta (f)\right ]+\frac {\partial }{\partial x_i \partial x_{\kern-1pt j}}\left [H(f) T_{\textit{ij}}\right ], \end{gathered} \end{align}
and
\begin{align} \left.\begin{array}{c} Q_j= \big[\rho (u_{\kern-1pt j}^{\prime }+U_{0 j}-v_{\kern-1pt j})+\rho _0 (v_{\kern-1pt j}-U_{0 j})\big], \\[5pt]L_{\textit{ij}}= \big[\rho u_{\kern-1pt i}^{\prime } (u_{\kern-1pt j}^{\prime }+U_{0 j}-v_{\kern-1pt j})+ (p-p_0) \delta _{\textit{ij}}-\tau _{\textit{ij}}\big], \\[5pt]T_{\textit{ij}}=\rho u_{\kern-1pt i}^{\prime } u_{\kern-1pt j}^{\prime }+ \big[ (p-p_0 )-c_0^2 (\rho -\rho _0)\big] \delta _{\textit{ij}}-\tau _{\textit{ij}}, \end{array} \right\} \end{align}
where
$Q_j$
and
$L_{\textit{ij}}$
are the source terms for thickness and loading noise, respectively;
$T_{\textit{ij}}$
denotes the quadrupole noise and can be ignored at a low Mach number;
$H$
and
$\delta$
are the Heaviside function and Dirac function, respectively;
$U_{0 j}$
is the free stream velocity;
$f$
is the penetrable surface,
$n_i$
is the unit normal vector of the integration face and
$(\bullet ^{\prime })$
denotes fluctuation components. The Green’s function is used to solve the FW–H equation, with the physical quantities on the integration surface obtained by ILES. The FW–H integration surface is generated by extruding a single profile along the spanwise direction. This profile consists of a semicircle, located
$10l_c$
from the trailing edge, and two attached straight lines with a length of
$20l_c$
. The position of the FW–H integration surface is shown by the red line in figure 1.
Due to computational cost limitations, the spanwise length in the simulation is shorter than that in the experiment; thus, Herr & Reichenberger (Reference Herr and Reichenberger2011) correction is performed and compared with the experimental data, expressed as
\begin{align} \textit{SPL}_{1 / 3({\textit{norm}})}=\textit{SPL}_{1 / 3}-50 \log \left (\textit{Ma}\right )-10 \log \big (\delta l_z / R_s^2\big ), \end{align}
where
$\delta$
is the characteristic outer length scale,
$l_z$
is the spanwise length and
$R_s$
is the distance from the observer and noise source.
2.3. The multi-process acoustic theory
The FW–H equation is capable of accurately predicting far-field noise; however, establishing a direct correlation between the nominal source terms in the FW–H equation and practical noise reduction remains challenging. The multi-process acoustic theory recently developed by Mao et al. (Reference Mao, Kang, Wu, Yu, Gao, Su and Lu2020) obtains a sound mode equation by decomposing the flow into vector transverse and scalar longitudinal processes. The sound mode equation characterises the flow noise sources based on the objective causality in the dynamic equations related to the longitudinal process, with its inhomogeneous convective form expressed as
\begin{align} D^2 \mathcal{R}-(D \mathcal{R})^2-\boldsymbol{\nabla }\boldsymbol{\cdot }\big (c^2 \boldsymbol{\nabla }\mathcal{R}\big )-\nu _\theta {\nabla} ^2 D \mathcal{R}=-\beta +\boldsymbol{\nabla }\boldsymbol{\cdot }\big (c^2 \boldsymbol{\nabla }s^*\big ), \end{align}
where
$\mathcal{R}=\ln (\rho / \rho _0)$
is the dimensionless acoustic variable,
$s^* = (s-s_0) / c_p$
is the relative entropy change,
$c_p$
is the specific heat at constant pressure,
$D$
denotes the material derivative,
$\nu _\theta =(\lambda +2 \mu ) / \rho$
represents the longitudinal kinematic viscosity and
$\lambda$
stands for the bulk viscosity. Here,
$\beta = (\vartheta \boldsymbol{I}-\boldsymbol{\nabla }\boldsymbol{u}^T ):\boldsymbol{\nabla }\boldsymbol {u}$
is considered as a pseudo ‘inviscid work rate’ done by the nonlinear inertial force over the boundary of the unit mass control volume;
$\vartheta$
and
$\boldsymbol{I}$
are the dilatation and the second-order unit tensor, respectively. At low-Mach-number conditions, the entropy term
$\boldsymbol{\nabla }\boldsymbol{\cdot }(c^2 \boldsymbol{\nabla }s^* )$
can be ignored owing to the limited temperature change, and the shear noise source (
$\beta$
) plays a dominant role. In the multi-process acoustic theory, the source terms have a clear physical interpretation and do not incorporate sound propagation effects. This characteristic facilitates the connection between changes in the physical source and far-field noise reduction. Additionally, the definition of the shear noise source is compact, and its computation can be directly derived from the spatial discretisation of the flow equations, making it advantageous and convenient for noise optimisation. A brief derivation of the sound mode equation is presented in figure 2, where
$h$
and
$\boldsymbol{\omega}$
denote enthalpy and vorticity, respectively.
A Bayesian optimisation framework based on the physical acoustic sources and FWH acoustic analogy.
2.4. Bayesian optimisation algorithm
In the present work, a parallel Bayesian optimisation algorithm (Wilson et al. Reference Wilson, Moriconi, Hutter and Deisenroth2017) is employed to optimise blowing or suction control in the vicinity of the aerofoil trailing edge, aiming to achieve the objective of reducing far-field noise. This algorithm can mitigate the risk of falling into local optima while simultaneously reducing the overall optimisation time. There are two major components of the Bayesian optimisation workflow: first, a surrogate model is employed to predict the objective function for unseen data points, and then an appropriate optimisation policy is used to determine the optimal location for the next samples.
Currently, the Gaussian process (GP) is selected as the surrogate model. Its predictions are provided in the form of probability distributions, enabling us to quantify the uncertainty in predictions and thereby better balance risk and reward in decision making. Specifically, suppose that the optimisation parameter space is
$m$
-dimensional and the objective function is denoted as
$f(\boldsymbol{x})$
. Initially, we create
$n$
points through a random number generator and compute their objective function values. Thus, the training dataset is
$\mathcal{D}:=\{X,\boldsymbol{f}\}$
, where
\begin{align} X:=\left [\begin{array}{ccc} \uparrow & & \uparrow \\ \boldsymbol{x}_1 & \ldots & \boldsymbol{x}_n \\ \downarrow & & \downarrow \end{array}\right ] \in \mathbb{R}^{m \times n}, \quad \boldsymbol{f}:=\left [\begin{array}{c} f\left (\boldsymbol{x}_1\right ) \\ \vdots \\ f\left (\boldsymbol{x}_n\right ) \end{array}\right ] \in \mathbb{R}^n.\end{align}
In this study,
$m = 2$
and
$n = 5$
. Subsequently, a GP model is trained on this dataset, enabling
\begin{align} f^* \mid X^*, \mathcal{D} \sim \mathcal{N}\left (\mu \left (X^*, \mathcal{D}\right ), \sigma \left (X^*, \mathcal{D}\right )\right ), \end{align}
where
$X^*$
is the test input and
$f^*$
represents the corresponding predicted values. Here
$\mu (X^*, \mathcal{D} )$
and
$\sigma (X^*, \mathcal{D} )$
are the mean and covariance matrix of a multivariate normal distribution, respectively, satisfying
\begin{align} \mu \left (X^*, \mathcal{D}\right )=\boldsymbol{K}\left (X^*, X\right ) \big(\boldsymbol{K}(X, X)+\sigma _n^2 \boldsymbol{I} \big)^{-1} \boldsymbol{f}, \\[-28pt] \nonumber \end{align}
\begin{align} \sigma \left (X^*, \mathcal{D}\right )=\boldsymbol{K}\left (X^*, X^*\right )-\boldsymbol{K}\left (X^*, X\right ) \big(\boldsymbol{K}(X, X) +\sigma _n^2 \boldsymbol{I}\big)^{-1} \boldsymbol{K}\left (X, X^*\right ), \\[2pt] \nonumber \end{align}
where
$\boldsymbol{K} (X, X^* )$
is a kernel matrix capturing the correlations between elements. To improve numerical stability, Gaussian noise
$\epsilon \sim \mathcal{N} (0,\sigma _n^2 )$
is added into the observations (Jenkins, Gerstoft & Park Reference Jenkins, Gerstoft and Park2023). The Matérn
$5/2$
kernel function (Stein Reference Stein1999; Matérn Reference Matérn2013) is employed in the current study, expressed as
\begin{align} \boldsymbol{K}\left (X, X^*\right ) = \sigma_{\kern-1.5pt f}^2 \left ( 1 + \sqrt {5}d \big(\boldsymbol{x}_i,\boldsymbol{x}_j^* \big) + \frac {5}{3}d^2 \big(\boldsymbol{x}_i,\boldsymbol{x}_j^* \big) \right ) \exp \big ( -\sqrt {5}d \big(\boldsymbol{x}_i,\boldsymbol{x}_j^* \big) \big ), \\[-28pt] \nonumber \end{align}
\begin{align} d \big(\boldsymbol{x}_i,\boldsymbol{x}_j^* \big) = \sqrt { \sum _{k=1}^m \dfrac {\big(x_{i,k} - x_{j,k}^* \big)^2}{\ell _k^2} }, \\[2pt] \nonumber \end{align}
where
$\sigma_{\kern-1.5pt f}^2$
denotes the covariance amplitude and
$\ell _k$
represents the length scale. The hyperparameters in GP are optimised by maximising the marginal likelihood, which automatically trades off the data fit against model complexity and thereby mitigates overfitting.
Following this, a Bayesian optimisation policy takes in the trained GP surrogate to identify the optimal location for the next samples through an acquisition function. Conventional Bayesian optimisation follows a sequential sampling strategy. In contrast, this work implements a parallel upper confidence bound (q-UCB) acquisition function to diminish the optimisation time (Wilson et al. Reference Wilson, Moriconi, Hutter and Deisenroth2017). The q-UCB trades off exploitation and exploration. In a q-UCB we need to simultaneously select q points, whose joint posterior distribution follows
$\mathcal{N} (\boldsymbol{\mu },\boldsymbol{\varSigma } )$
. Here
$\boldsymbol{\mu } \in \mathbb{R}^{q}$
is the mean vector and
$\boldsymbol{\varSigma } \in \mathbb{R}^{q\times q}$
is the covariance matrix. Specifically, a q-UCB can be expressed as
\begin{align} q-\mathrm{UCB}(\boldsymbol{x} ; \kappa )=\int _{-\infty }^{\infty } \max (\boldsymbol{\mu }+|\hat {\boldsymbol{L}} \boldsymbol{z}|) \mathcal{N}(\boldsymbol{z} ; \boldsymbol{0}, \boldsymbol{I}) {\rm d} \boldsymbol{z}, \end{align}
where
$\hat {\boldsymbol{L}} \hat {\boldsymbol{L}}^{\top }=\tilde {\boldsymbol{\varSigma }}=({\kappa \pi }/{2}) \boldsymbol{\varSigma }$
. The above integral can be approximated using the Monte Carlo method:
\begin{align} q-\mathrm{UCB}(\boldsymbol{x} ; \kappa ) \approx \frac {1}{p} \sum _{k=1}^p \max (\boldsymbol{\mu }+ |\hat {\boldsymbol{L}} \boldsymbol{z}_k | ), \quad \boldsymbol{z}_k \sim \mathcal{N}(\boldsymbol{0}, \boldsymbol{I}). \end{align}
In the above acquisition function,
$\kappa$
serves as a pivotal hyperparameter. Larger
$\kappa$
values bias the strategy toward exploration of uncertain regions, while smaller
$\kappa$
values favour exploitation of known optima. In this study,
$q = 5$
,
$p = 512$
and
$\kappa = 2$
. Note that the q-UCB method focuses on the maximum value in a batch of data rather than the average performance of the batch.
Two rounds of Bayesian optimisation were conducted. In the first round, the far-field noise was directly optimised; in the second round, near-field acoustic sources were utilised to construct the GP model. All other optimisation parameters remained unchanged. A schematic diagram of the aforementioned Bayesian optimisation algorithms is presented in figure 2. The iteration can be considered converged when the historical optimum of the objective function falls within the statistical error margin of the best value for five consecutive iterations.
3. The configurations of the numerical model
3.1. The computational schemes and meshes
Similar to the numerical simulation conditions conducted by Hu et al. (Reference Hu, Wan, Ye, Sun and Lu2022), a NACA0012 aerofoil is placed in a uniform flow at
$Ma = 0.1631$
. The zig–zag strips are installed on both sides of the aerofoil to ensure a complete boundary layer transition. The detailed simulation parameters are presented in table 1. A schematic illustration of the computational domain is depicted in figure 1(a). The origin of the Cartesian coordinate system is placed at the aerofoil leading edge, with the
$x$
axis aligned with the free-stream direction, the
$y$
axis normal to the chord and the
$z$
axis coinciding with the spanwise direction.
The details for flow simulations. Here
$Ma$
is the free-stream Mach number,
$Re_c$
denotes the Reynolds number based on the aerofoil chord length,
$l_z$
is the non-dimensional spanwise length,
$\phi$
is the angle of attack, and
$x_\Delta$
,
$h_\Delta$
,
$l_\Delta$
, and
$\lambda _\Delta$
represent the position, height, width and spanwise wavelength of the transition strip, respectively;
${\rm d}t$
is the dimensionless time step, DoF is the degree of freedom of the mesh and
$b$
and
$C_\mu$
are the streamwise length and intensity of the flow control.
An adiabatic no-slip boundary condition is applied at the solid wall, while the periodic boundary condition is imposed on the spanwise direction. The Riemann condition is specified at the far field, and steady blowing/suction is employed at the control slots. In this study, both the length of the control slot
$b$
and the flow control intensity
$C_\mu$
are held constant at
$0.05l_c$
and
$0.8$
, respectively. The definition of flow control intensity is consistent with that of Szőke et al. (Reference Szőke, Fiscaletti and Azarpeyvand2020a
), i.e.
$C_\mu = {u_c b}/{u_0 \theta _0}$
, where
$u_c \approx 0.09$
and
$\theta _0 \approx 0.0056l_c$
denote the blowing/suction velocity and the momentum thickness of the uncontrolled boundary layer near the trailing edge. Notably, in the experiment by Szőke et al. (Reference Szőke, Fiscaletti and Azarpeyvand2020a
), the suction intensity ranged from 1.8 to 9.1. However, the present study aims to achieve optimal control effectiveness with minimal energy consumption, so the flow control intensity is significantly lower than the values used in that experiment. The objective of this work is to achieve maximum noise reduction under low flow control intensity, where the centre position
$x/l_c$
and angle of blowing/suction
$\alpha$
are defined as parameters to be optimised. The control slot is positioned near the trailing edge, with its centre location
$x/l_c$
varying from
$0.825$
to
$0.975$
in the chordwise position, corresponding to a potential coverage domain of
$[0.8, 1.0]$
. The potential installation region for the control slot is illustrated by the red surface in figure 1(b). The active control angle domain spans
$[0, 2\pi ]$
, where angles in
$[0, \pi ]$
denote suction and those in
$[\pi , 2\pi ]$
represent blowing. The definition of the blowing/suction angle is illustrated in figure 1(d).
All controlled cases are computed by restarting from the solution of the uncontrolled case. The dimensionless time step and sampling time interval are
$8\times 10^{-5}$
and
$1\times 10^{-2}$
, respectively. The duration to eliminate the restart transient is
$25l_c/U_0$
, and the noise statistics are collected for 200 dimensionless times (about 32 flow-past times). The fluctuation signal is partitioned into four segments with
$50\,\%$
overlap between adjacent sections, and a Hanning window is applied to minimise spectrum leakage.
The same grid is adopted for all cases. The enlarged view of the mesh topology is presented in figure 1(b–d). Hexahedral elements are implemented in the near-wall and near-wake regions with a stretching ratio of
$1.05$
, while tetrahedral elements populate the vicinity of the transition strip and far-field domains. The minimum wall distance
$\Delta y^+$
of the solution point is maintained below 1 to fully resolve the viscous sublayer. The total degree of freedom (DoF) is approximately
$3 \times 10^7$
, distributed along the aerofoil span with
$80$
solution points.
3.2. Validation
For validation purposes, the results of the uncontrolled case are compared against previous numerical (Hu et al. Reference Hu, Wan, Ye, Sun and Lu2022) and experimental data (Brooks et al. Reference Brooks, Pope and Marcolini1989; Lee & Kang Reference Lee and Kang2000). Figure 3 presents the mean pressure coefficient on the wall, the skin friction coefficient in the turbulent boundary layer region and the normalised sound pressure level (SPL) in 1/3-octave bands at the probe of
$(x/l_c,y/l_c,z/l_c)=(1.0, 48.03, 0.05)$
. The present results show good agreement with previous numerical and experimental data, indicating the reliability of the current study. The sudden change in the pressure coefficient in figure 3(a) is due to the transition strip applied in the numerical model. There is a small area of inconsistency between the present results and Hu et al. (Reference Hu, Wan, Ye, Sun and Lu2022)’s results after the transition trip. This is likely due to the numerical overshoot caused by the transition from tetrahedral to hexahedral elements at approximately
$x/l_c \approx 0.25$
in the current study. Since this region is located far from the areas of noise generation and control application, it does not have a fundamental impact on the optimisation results presented in this research. The differences between the numerical and experimental SPL in figure 3(c) should result from the difference in turbulence tripping methods and the unresolved small scales. In figure 3(c) we also provide the noise prediction results obtained using an integration surface with a closing cap and a computational domain with doubled spanwise extent. It can be observed that selecting different integration surfaces or increasing the spanwise length does not significantly alter the current results.
Comparisons between the present ILES data and previous studies (Brooks, Pope & Marcolini Reference Brooks, Pope and Marcolini1989; Lee & Kang Reference Lee and Kang2000; Hu et al. Reference Hu, Wan, Ye, Sun and Lu2022): (a) mean pressure coefficient distribution on the surface of the aerofoil, (b) the skin friction coefficient in the turbulent boundary layer region, and (c) the normalised sound pressure level (SPL) in 1/3-octave bands. The observer is located at
$(x/l_c,y/l_c,z/l_c) = (1.0, 48.03, 0.05)$
, and
$St_\delta = f \delta _{99}/U_c$
is the Strouhal number based on the boundary layer thickness and convection velocity. The turbulent boundary layer thickness
$\delta _{99}/l_c$
in the current study, Hu et al. (2022)’s numerical simulation and Brooks et al. (1989)’s experiment are
$0.0426$
,
$0.0426$
, and
$0.0753$
, respectively.
4. Results and discussion
4.1. The results of Bayesian optimisation
Bayesian optimisation is carried out to determine the optimal parameters for active control, specifically the centre position of slot
$x/l_c$
and the blowing/suction angle
$\alpha$
, for noise reduction. Naturally, the change in the far-field overall SPL (
$\text{OASPL}$
) is chosen as the metric for assessing noise reduction. In this direct optimisation process the mean overall sound pressure reduction level (
$\overline {\Delta \text{OASPL}}$
) over the direction range from about
$40^{\circ }$
to
$140^{\circ }$
is designated as the objective function. The observers are distributed along a circle as indicated by the orange dots in figure 1(a). The angular separation between adjacent monitors is
$6^{\circ }$
. As shown in figure 4, a GP model is employed to compute the posterior distribution of the objective function, including the mean and uncertainty of the noise reduction level. The subsequent sampling location is determined via the q-UCB acquisition policy. Here, the uncertainty in the data is quantified by the variance predicted by the surrogate model and normalised by the maximum variance across all prediction points. Thus, in the right panel of each subgraph, a value of 1 represents a high level of prediction uncertainty, while a value of 0 indicates that the prediction is already quite accurate. The uncertainty drops sharply to zero at the sampling locations, resulting in distinct minima in the visualisation. In the left panel of each subgraph, all regions predicted to have increased noise are displayed in blue. Additionally, we use white points to indicate the location for the next sampling batch guided by the q-UCB strategy. In this study, one iteration step updates five sampling points simultaneously.
Progress made by the Bayesian optimisation. The GP predictions at (a) iteration 0, (b) iteration 1, (c) iteration 2, (d) iteration 3, (e) iteration 5, (f) iteration 10, (g) iteration 11 and (h) iteration 12 are displayed. In each subplot, the left and right panels display the
$\overline {\Delta \text{OASPL}}$
values and the uncertainty of the data predicted by the 2D GP model, respectively.
Figure 4 shows the prediction results of some iteration steps in the first direct Bayesian optimisation process in detail. As shown in figure 4(a), we generated the initial dataset using random numbers. The sampling point locations are in the blue parts of the right panel. The GP model generated from the initial dataset in the left panel identifies a promising region within the parameter space near the sampling point at the bottom-right corner, where noise reduction is anticipated. Subsequent sampling locations are strategically concentrated around this region. Following the first iteration, as depicted in figure 4(b), the noise reduction region is updated, with the predicted next sampling location shifting leftward (toward the trailing edge upstream). By the second iteration, both the predicted noise reduction region and the next sampling location further move upstream. Meanwhile, a data point demonstrating over
$3.5$
dB noise reduction has emerged in the current dataset. After three iterations, the predicted noise reduction trend has taken initial shape. In the fifth iteration (figure 4
e) we identify the optimal parameters for this round of optimisation. The optimal control parameters include a position located at
$x/l_c=0.8734$
and an angle of
$0.5895\pi$
, which can achieve a noise reduction of up to
$3.7$
dB. In the experiments on parametric suction control at high Reynolds numbers (
$Re = 10^6$
) by Szőke et al. (Reference Szőke, Fiscaletti and Azarpeyvand2020a
), the optimal suction angle was
$0.61\pi$
(note that the angle definition in the experiment differs from that in this paper), which is very close to the optimal parameter of
$0.5895\pi$
in this study. Note that optimal control incurs a certain drag penalty, increasing the drag coefficient by about
$22\,\%$
. This implies a conflict between noise reduction effectiveness and aerodynamic performance. After 10 iterations, as illustrated in figure 4(f–h), the GP model begins predicting multiple local maxima in noise reduction while the global maximum remains unchanged. At this stage, the q-UCB acquisition strategy exhibits a distinctive behavioural pattern: within each batch of sampling points, one or two consistently localise near existing extrema while others probe unexplored regions of the parameter space. This indicates that the q-UCB scheme balances exploration and exploitation while avoiding getting trapped in local optima. After sufficient iterations, the uncertainty value corresponding to the noise reduction region has become relatively small. Therefore, we can consider that the predictions provided by the GP model are tending towards accuracy. The black solid line in figure 5 further shows the history of the objective function for our first Bayesian optimisation. Considering a statistical error of
$\pm 0.1$
dB, we can regard the current iteration as having converged, with the optimal value found at the fifth iteration.
The optimisation history of the
$\overline {\Delta \text{OASPL}}$
(solid black line) and
$\beta ^*$
(solid red line). The black shaded area indicates the error range of
$\pm 0.1$
dB and the red shaded area represents the error range of
$\pm 0.05$
. The black pentagram marks the optimum noise reduction value based on GP models constructed by
$\overline {\Delta \text{OASPL}}$
, and the red pentagram indicates the maximum noise reduction corresponding to GP models constructed by
$\beta ^*$
.
The final GP model is built from a total of 80 high-fidelity samples and is displayed in figure 6(a). The ultimate surrogate model exhibits several noteworthy findings. Under the current simulation set-up, all blowing controls produce the result of noise enhancement, corresponding to the region where
$\alpha /\pi \gt 1$
in figure 6(a). Notably, suction controls positioned close to the trailing edge fail to yield effective control outcomes. However, when we shift the suction slot upstream, normal suction demonstrates improved performance. As the suction slot continues to move further upstream, the effective range of suction angles gradually expands. Nevertheless, when the suction slot is positioned significantly away from the trailing edge, the efficacy of suction control slightly diminishes. Moreover, the GP model in figure 6(a) exhibits multiple local extrema, with the optimal control parameters identified through our iterative process marked by black pentagrams.
Comparison of GP models based on (a) far-field noise reduction
$\overline {\Delta \text{OASPL}}$
, and (b) near-field physical noise source strength
$-\beta ^*$
. Both GP models utilise the same full set of 80 sample points from the first round of Bayesian optimisation. The black pentagram represents the optimal noise reduction point, while magenta diamonds denote points used for comparison in subsequent analysis. The white square denotes the optimal parameters identified during the second iteration of the source-oriented Bayesian optimisation.
It should be emphasised that although Bayesian optimisation efficiently identifies optimal control parameters, its computational cost remains prohibitively high for 3D aeroacoustic applications. Acoustic simulations typically demand high fidelity, extended statistical sampling duration and incur computational overhead from far-field signal delays, collectively making each sampling iteration prohibitively expensive. Taking the aforementioned optimisation procedure as an example, it consumed approximately
$1\times 10^5$
DCU-hours in total by utilising 20 DCU cards over a period of more than seven months. Consequently, we attempt to integrate multi-process acoustic theory into the Bayesian optimisation framework to circumvent direct far-field noise computation, thereby achieving significant computational savings. In this study we define a near-field noise source strength
$\beta ^*$
defined in the multi-process acoustic equation (Mao et al. Reference Mao, Kang, Wu, Yu, Gao, Su and Lu2020), formulated analogously to the overall SPL. Here
$\beta ^*$
is given by
\begin{align} \beta ^* = 10\log _{10}\left (\frac {\int \overline {\beta ^2} {\rm d} V}{\int \overline {\beta _0^2} {\rm d} V}\right )\!, \end{align}
where
$\beta$
and
$\beta _0$
denote the shear noise source with and without control, respectively. Here
$(\bar {\bullet })$
represents the time average;
$V$
denotes the domain for physical source integration, selected to encompass main trailing-edge noise scattering regions and also include partial wake structures, i.e.
$V\in [l_c-2\delta _0,l_c+\delta _0] \times [-1.5\delta _0,1.5\delta _0]\times [0,0.1l_c]$
(
$\delta _0$
is the boundary layer thickness for the uncontrolled case). A positive value of
$-\beta ^*$
indicates a decrease in sound source intensity, whereas a negative value signifies an increase.
Before the second round of Bayesian optimisation, we first examine whether the GP model constructed from the near-field physical source intensity can capture the trend of noise reduction. In figure 6(b) the dataset for the surrogate model comprises all 80 ILES results obtained by the previous optimisation process. Notably, the parameter space with increased source intensity is shaded blue. Comparing the far-field noise reduction GP model (i.e. figure 6
a) with the physical source intensity GP model (figure 6
b) reveals some interesting phenomena. First, all blowing-control cases exhibit increased noise source intensity. Second, suction slots adjacent to the trailing edge show less significant source intensity reduction. As the control slot shifts upstream, normal suction initially demonstrates favourable sound source attenuation, followed by an expansion of the effective suction angle range. Finally, the regions exhibiting significant attenuation of sound source intensity broadly coincide with those achieving substantial noise reduction. All the above findings align consistently with the trends observed in far-field noise variation. It should also be mentioned that the GP model of
$\beta ^*$
exhibits inadequate resolution in regions where
$\overline {\Delta \text{OASPL}} \gt 3.3$
dB, failing to resolve multiple local extremes in figure 6(a). In general, the selection of
$\beta ^*$
for constructing the surrogate model should have effectively captured the dominant physical mechanisms, which is crucial for its application in coarse-grained optimisation.
It is worth emphasising that selecting
$\beta ^*$
as the objective function offers additional advantages. Figure 7(a) compares the time delay between far-field pressure fluctuations and near-field noise sources. The blue pressure signal is recorded at the FW–H surface directly above the trailing edge, while the green pressure signal is measured at a position further downstream. Here,
$t = 0$
denotes the moment when control is activated, i.e. the restart moment. It can be seen that the far-field monitors detect the effects of control after a long time delay: the blue signal responds at
$t \approx 10$
(1.6 flow-past times) and the green signal at
$t \approx 20$
(3.3 flow-past times). Moreover, the requisite waiting period for signal stabilisation on the FW–H surface is considerably longer, necessitating at least 90 non-dimensional time units (14.7 flow-past times) before initiating noise statistics. In contrast,
$\beta ^*$
responds to the control almost instantaneously (
$t \approx 0$
), enabling us to start a statistical analysis of sound source intensity as early as
$t \approx 15$
(2.4 flow-past times). Figure 7(b) further compares the statistical convergence of
$\overline {\Delta \text{OASPL}}$
and
$\beta ^*$
, where the relative error is defined as
\begin{align} \text{Relative error} = \left |\frac {q_{{t}}-q_{{\textit{ref}}}}{q_{{\textit{ref}}}}\right |, \end{align}
where
$q$
denotes
$\overline {\Delta \text{OASPL}}$
or
$\beta ^*$
. We consider the value obtained from sufficiently long-duration statistics (
$t=[150,550]$
, approximately
$65$
flow-past times) as the reference value
$q_{{{ref}}}$
. Here
$q_t$
represents the value accumulated up to time
$t$
, where
$t=150$
is the moment when the statistics start. Evidently,
$\beta ^*$
converges more rapidly than
$\overline {\Delta \text{OASPL}}$
. The current noise statistics satisfy an error tolerance below
$0.1$
dB, corresponding to a relative error
${\lt } 0.027$
. Attaining improved statistical convergence would necessitate a substantially extended sampling duration. In contrast, achieving a noise source intensity statistical error below
$0.1$
(relative error
${\lt } 0.02$
) requires less than 10 non-dimensional times. Additionally, the computation of shear sound sources imposes no additional computational overhead on ILES, as these values can be obtained directly from the spatial discretisation of the governing equations. In short, the combination of robust physical representation, reduced signal latency and faster statistical convergence enables Bayesian optimisation based on physical noise sources to substantially reduce computational costs.
Comparison of costs between statistical far-field noise reduction and near-field physical noise sources for the optimal case: (a) the signal delay of pressure fluctuations on the FWH face and near-field physical noise source strength, and (b) the statistical convergence of
$\overline {\Delta \text{OASPL}}$
and
$\beta ^*$
.
Next, we examine the effects of the sound-source-oriented coarse-grained Bayesian optimisation. During the coarse-grained optimisation process,
$\beta ^*$
is selected to construct the GP model, with the initial dataset consistent with the first round of optimisation. In this optimisation, 75 dimensionless time units are used to eliminate transient effects, followed by an additional 75 units for source statistics. All other settings remain identical to the first optimisation round. Under the current statistical approach, the single sampling cost of source-oriented Bayesian optimisation is approximately
$40\,\%$
of that for direct Bayesian optimisation. However, this statistical estimation is highly conservative. If we adopt a less conservative statistical method, i.e. initiating measurement immediately upon signal stabilisation while maintaining identical statistical error thresholds (
$0.1$
dB for
$\overline {\Delta \text{OASPL}}$
and
$0.1$
for
$\beta ^*$
), the single sampling cost of coarse-grained optimisation would thus be reduced to approximately
$10\,\%$
. The history of this optimisation is shown as a red solid line in figure 5. Within the margin of statistical error, parameters minimising source strength were identified as early as the second iteration. Additionally, we calculate the maximum noise reduction for samples in the second iteration step (statistical analysis over 200 dimensionless time units), as indicated by the red pentagram. Notably, the use of physical source intensity enables faster identification of optimal parameters. Under the current statistical strategy, the comprehensive cost has been reduced to about one-sixth of the original. The cost calculation here is based on a reduction in per-sample cost to
$40\,\%$
, with the number of iterations required to find the optimum or achieve convergence also reduced to
$40\,\%$
of the original, without accounting for the cost of generating the initial dataset. It should be noted that while different initial datasets can influence the iterative process, using a physical source is likely to converge faster due to the smoother surrogate model of
$\beta ^*$
, as shown in figure 6(b). Additionally, the white square in figure 6(a) shows optimal parameters from the second iteration in the source-oriented optimisation. As illustrated, the coarse-grained optimisation identifies an alternative extremum from the perspective of sources. The control effect approaches the global optimum achieved through direct Bayesian optimisation, while the control parameters remain close to their counterparts in the first round of optimisation. The discrepancy between the optimal parameters obtained from coarse-grained optimisation and the global optimum should be mainly attributed to the measurement method of the source term, including the selected spatial region and the defined objective function of the source term. Overall, the source-oriented method demonstrates the effectiveness of the optimisation process and can be directly deployed for noise control applications. Additionally, it can also be used to rapidly constrain the search domain to facilitate locating the global optimum.
4.2. Noise emission and flow structures
In this section we focus on discussing the changes in far-field noise and near-field flow structures induced by blowing and suction control. To this end, we have selected the uncontrolled case for comparison with the optimally controlled case (X087A059). Furthermore, two additional cases are included: one featuring a different suction angle (X087A017) and another with a different suction location (X093A059). Additionally, one blowing case (X087A150) exhibiting a noise intensification effect is selected as a reference to facilitate a more comprehensive examination of the source mechanism. These points used for analysis are indicated by the markers in figure 6(b), with the detailed parameter settings given in table 2.
The cases used for noise and flow structure analysis. The uncontrolled case is selected as the baseline case.
Figure 8 illustrates the distribution of far-field pressure fluctuations across different frequency bands, normalised by the maximum value along the circle for the uncontrolled case depicted in figure 8(a). Subsequently, we define the low-frequency band as
$St = [2, 6]$
, the mid-frequency band as
$St = [6, 14]$
and the high-frequency band as
$St = [14, 20]$
. For the blowing case, different ticks are used for the whole-frequency band and low-frequency band, as labelled by the magenta ticks. It can be observed from profiles of
$\sqrt { \langle p^{\prime 2} \rangle }$
that suction control has no significant effect on altering the noise propagation direction across different frequency bands. In contrast, blowing control alters the primary radiation direction of low-frequency noise, thereby influencing the overall radiation pattern across the entire frequency band. Specifically, the dominant acoustic propagation direction is about
$132^{\circ }$
for both the uncontrolled and suction cases in figure 8(a), whereas it shifts to about
$114^{\circ }$
for the blowing case. At low frequencies, all suction control cases achieve significant reductions in far-field pressure fluctuations. Two dominant lobes are observed, suggesting the presence of compact dipole sources near the trailing edge. Among these, the optimal case X087A059 exhibits the most effective low-frequency noise reduction capability. Conversely, the blowing-control case X087A150 substantially increases the amplitude of low-frequency pressure fluctuations. In the mid-frequency range, all cases achieve a certain degree of noise reduction. Similarly, case X087A059 demonstrates the best control performance. Additionally, some non-impact properties occur accompanied by four lobes pointing to
$\theta \approx \pm 138^{\circ }$
and
$\theta \approx \pm 90^{\circ }$
. At high frequencies, the distribution of far-field pressure fluctuations becomes increasingly irregular. Among the selected cases, the blowing case X087A150 demonstrates the most effective control performance, while the noise reduction capabilities of the suction cases remain relatively similar.
Directivity plot of the acoustic intensity
$\sqrt {\langle p^{\prime 2}\rangle }$
on a circle with radius
$R=48.03l_c$
in various frequency ranges, with the Strouhal number defined as
$St = fl_c/U_0$
:
$(a)$
whole-frequency band,
$St = [0, 20]$
,
$(b)$
low-frequency band,
$St = [2, 6)$
,
$(c)$
mid-frequency band,
$St = [6, 14)$
, and
$(d)$
high-frequency band,
$St = [14, 20]$
.
Figure 9 displays the acoustic propagation patterns and instantaneous vortex structures for the selected cases. From the dilatation field, it can be seen that the sound waves are generated by scattering at the aerofoil trailing edge. Suction control attenuates acoustic wave intensity, whereas blowing significantly amplifies SPLs with a pronounced increase in long-wavelength components. The isosurfaces of
$Q$
in figure 9(d) demonstrate that blowing induces additional large-scale vortex shedding in the wake, thereby triggering low-frequency noise enhancement. In contrast, suction control primarily narrows the core separation of wake vortices without substantially altering the flow pattern. This observation aligns with the unchanged noise radiation directivity of the suction cases in figure 8.
Instantaneous sound propagation and vortex structures shown by the dilatation
$\vartheta$
and isocontours of the Q-criterion (
$Q = 0.1$
coloured by streamwise velocity): (a) the uncontrolled case, (b) the optimal case X087A059, (c) the suction case X087A017, and (d) the blowing case X087A150.
Dynamic mode decomposition (DMD) (Schmid Reference Schmid2010) is further employed to clarify the effects of blowing and suction controls on flow structures. A sequence of continuous pressure fluctuation field snapshots is assembled into matrices
$\boldsymbol{P}_1^{N-1}$
and
$\boldsymbol{P}_2^N$
, i.e.
\begin{align} \boldsymbol{P}_1^{N-1}=\boldsymbol{P}_{\boldsymbol{x}, t(1 \ldots N-1)}=\left [\begin{array}{cccc} p^{\prime }\left (\boldsymbol{x}_1, t_1\right ) & p^{\prime }\left (\boldsymbol{x}_1, t_2\right ) & \ldots & p^{\prime }\left (\boldsymbol{x}_1, t_{N-1}\right ) \\ p^{\prime }\left (\boldsymbol{x}_2, t_1\right ) & p^{\prime }\left (\boldsymbol{x}_2, t_2\right ) & \ldots & p^{\prime }\left (\boldsymbol{x}_2, t_{N-1}\right ) \\ \vdots & \vdots & \ddots & \vdots \\ p^{\prime }\left (\boldsymbol{x}_i, t_1\right ) & p^{\prime }\left (\boldsymbol{x}_i, t_2\right ) & \ldots & p^{\prime }\left (\boldsymbol{x}_i, t_{N-1}\right ) \end{array}\right ]\!, \end{align}
\begin{align} \boldsymbol{P}_2^N=\boldsymbol{P}_{\boldsymbol{x}, t(2 \ldots N)}=\left [\begin{array}{cccc} p^{\prime }\left (\boldsymbol{x}_1, t_2\right ) & p^{\prime }\left (\boldsymbol{x}_1, t_3\right ) & \ldots & p^{\prime }\left (\boldsymbol{x}_1, t_N\right ) \\ p^{\prime }\left (\boldsymbol{x}_2, t_2\right ) & p^{\prime }\left (\boldsymbol{x}_2, t_3\right ) & \ldots & p^{\prime }\left (\boldsymbol{x}_2, t_N\right ) \\ \vdots & \vdots & \ddots & \vdots \\ p^{\prime }\left (\boldsymbol{x}_i, t_2\right ) & p^{\prime }\left (\boldsymbol{x}_i, t_3\right ) & \ldots & p^{\prime }\left (\boldsymbol{x}_i, t_N\right ) \end{array}\right ] \!, \end{align}
where
$\boldsymbol{P}_1^{N-1}$
and
$\boldsymbol{P}_2^N$
represent the matrices corresponding to time steps
$1$
to
$N-1$
and
$2$
to
$N$
, respectively. The time interval is taken as
$\Delta t = 0.1 l_c /c_0$
and the number of snapshots is
$500$
.
$\boldsymbol{x}$
contains all solution points excluding the sponge zone. Assume that there exists a linear mapping operator
$\boldsymbol{A}$
describing the dynamic characteristics of the flow system, that is,
\begin{align} \boldsymbol{P}_2^N=\boldsymbol{A} \boldsymbol{P}_1^{N-1}=\boldsymbol{P}_1^{N-1} \boldsymbol{S}+\boldsymbol{r} \boldsymbol{e}_{N-1}^T, \end{align}
where
$\boldsymbol{S}$
is the adjoint operator,
$\boldsymbol{r}$
denotes the residual and
$\boldsymbol{e}_{N-1}^T$
represents the
$(N-1)$
th vector of the identity matrix. The eigen-decomposition of the adjoint operator
$\boldsymbol{S}$
can be expressed as
\begin{align} \boldsymbol{S} \boldsymbol{q} = \lambda \boldsymbol{q}. \end{align}
The eigenvalue
$\lambda$
and eigenvector
$\boldsymbol{q}$
can be obtained via the singular value decomposition. The DMD modes are denoted by
$\boldsymbol{\phi }_i = \boldsymbol{L} \boldsymbol{q}_i$
, with the corresponding mode frequencies being
$St = \mathrm{Im}(\mathrm{log}(\lambda ))l_c/(2\pi U_0 \Delta t)$
. Here,
$\boldsymbol{L}$
represents the left singular matrix and
$\mathrm{Im}$
denotes the imaginary part operator. The energy of the pressure mode denoted
$E_{\phi _i}$
is defined as the
$L_2$
-norm of the modal vector. For comparison purposes, the energy ratio is defined as
\begin{align} Er_j=\sum _{St_{\phi _i} \in [j-1,j]} E_{\phi _i}/E_0, \quad j = 1, 2, 3, \ldots , 20, \end{align}
where
$E_0$
is the sum of mode energies in the most energetic frequency band (
$St=[5,6]$
) for the uncontrolled case. Figure 10 presents the distribution of mode energy ratios across different frequency bands for various cases. Note that the blowing case X087A150 employs distinct tick marks. For the uncontrolled case, and all suction cases, i.e. X087A059, X087A017 and X093A059, the
$St=[5,6]$
band constitutes the most energetic frequency range, which is correlated with hydrodynamically unstable waves exhibiting the most pronounced convective amplification. Assuming a convection velocity of
$U_c = 0.7U_0$
, the corresponding characteristic spatial scale
$\lambda _{ch} = U_c/f$
for this band ranges from
$0.117l_c$
to
$0.140l_c$
. In contrast, the most energetic frequency for case X087A150 occurs at
$St \approx 2.85$
. Here, the dominant frequency band (
$St = [2,3]$
) is associated with blowing-induced large-scale vortex shedding. The wavelength of these large-scale vortices is estimated as
$\lambda _{vs} = U_c l_c / (2.85U_0) \approx 0.25l_c$
, aligning with both the measured shedding wavelength in figure 9(d) and the mode structure wavelength in figure 11(d). Furthermore, compared with the uncontrolled case, blowing elevates the overall mode energy, while suction effectively attenuates mode energy across most frequency bands. Notably, the optimal control case X087A059 demonstrates the superior performance, achieving a significant reduction in mode energy throughout all frequency bands. Next, the analysis is conducted for modes with the most energy at two typical frequencies, i.e.
$St \approx 2.5$
and
$St \approx 5.5$
, as indicated by black arrows in figure 10.
Distribution of DMD mode energy ratios across various frequency bands. Here, the left black y ticks on the left are adopted for the baseline and suction cases, and the magenta y ticks on the right are used for the blowing case X087A150.
The slice view (
$z = 0.05l_c$
) of the real part of the most energetic pressure mode at
$St \approx 2.5$
for various cases and the corresponding isocontours of the mode structures: (a) the uncontrolled case, (b) the optimal case X087A059, (c) the suction case X087A017, and (d) the blowing case X087A150.
Figure 11 presents the DMD modes corresponding to the frequency
$St \approx 2.5$
. It can be observed that the baseline case exhibits regular near-field flow structures. The suction control can disrupt this regular pattern. In the optimal control case X087A059, the flow structure in the wake has become barely discernible in figure 11(b). The near-field structures in the less effective suction case X087A017 still exhibit weaker wake structures in figure 11(c). However, the blowing cases exhibit an entirely distinct flow mode – specifically, antisymmetric mode structures associated with large-scale vortex shedding. The radiated pressure fluctuations also correlate with changes in the near-field mode patterns. Among these, case X087A059 exhibits the weakest radiated noise, followed by the less effective suction case X087A017, while the blowing case X087A150 shows significantly enhanced noise radiation. Figure 12 displays the most energetic mode around the frequency
$St \approx 5.5$
. Similar to the low-frequency mode, the radiated noise decreases in the suction cases, while it increases in the blowing cases. The control effect is most effective in the optimal case X087A059. At current mode frequencies, the dominant radiation directions are pointing toward the upstream. The near-field flow structure also undergoes significant changes. Compared with the mode at
$St \approx 2.5$
, the flow structures in figure 12 exhibit shorter wavelengths. Both suction control and blowing control induce structural distortion in the wake. Specifically, the optimal suction control (X087A059) causes rapid dissipation of flow structures, whereas the blowing control (X087A150) results in stronger wake structures that persist for longer distances.
The slice view (
$z = 0.05l_c$
) of the real part of the most energetic pressure mode at
$St \approx 5.5$
for various cases and the corresponding isocontours of the mode structures: (a) the uncontrolled case, (b) the optimal case X087A059, (c) the suction case X087A017, and (d) the blowing case X087A150.
To quantitatively characterise variations in turbulent structures across different regions of the trailing-edge boundary layer, the anisotropy invariant map method is employed (Pope Reference Pope2001; Simonsen & Krogstad Reference Simonsen and Krogstad2005). In the case of negligible compressibility, the anisotropic part of the Reynolds stress tensor is expressed as follows:
\begin{align} b_{\textit{ij}}=\frac {\big \langle u_{\kern-1pt i}^{\prime } u_{\kern-1pt j}^{\prime }\big \rangle }{\left \langle u_k^{\prime } u_k^{\prime }\right \rangle }-\frac {1}{3} \delta _{\textit{ij}}. \end{align}
The second and third principal invariants
$I_2$
and
$I_3$
can be determined through the roots of the corresponding characteristic polynomial, namely,
\begin{align} \begin{gathered} I_2 = - \frac {1}{2} b_{\textit{ij}} b_{\textit{ji}},\\ I_3 = \det ([b_{\textit{ij}}]). \end{gathered} \end{align}
Lumley & Newman (Reference Lumley and Newman1977) argued that these invariants provide a complete description of turbulence anisotropy characteristics. To gain immediate visual insight, we select two independent invariants for turbulent stress invariance analysis, defined as follows (Pope Reference Pope2001):
\begin{align} \begin{gathered} 6 \eta ^2=-2 I_2=b_{\textit{ij}} b_{\textit{ji}},\\ 2 \xi ^3=I_3=\operatorname {det} ( [b_{\textit{ij}}]). \end{gathered} \end{align}
Figure 13 displays the anisotropy invariant map with different wall-normal distances at
$x=0.96l_c$
, where turbulence regimes delimited by black dashed boundaries are annotated in the legend. Active flow control is shown to modify near-wall turbulence structures at the trailing edge significantly. As can be seen from figure 13(a–b), the viscous sublayer (
$y^+ \lesssim 5$
) corresponds to the region leftward of the upper-right inflection point in the anisotropy invariant map for the uncontrolled case and the two upstream suction cases (X087A059, X087A017). Flow states within this domain align more closely with the two-component turbulence regime defined by curve I. Notably, the inflection points for cases X087A059 and X087A017 shift toward the single-component turbulence limit. In the buffer layer (
$5 \lesssim y^+ \lesssim 30$
), turbulence in the uncontrolled case approaches a rod-like axisymmetric state. Case X087A017 shows a slight deviation from this state, while the optimal control case (X087A059) exhibits a more significant departure. In figure 13(c), suction control near the trailing edge (X093A059) exhibits distinct behaviour: Its near-wall region (
$y^+ \lesssim 30$
) shows flow structures evolving toward disk-like turbulence (curve III). In the logarithmic layer, turbulence states for the suction cases gradually revert to uncontrolled conditions (see figure 13
a–c). Among these, the optimal suction case (X087A059) maintains noticeable modifications within the logarithmic layer. At higher boundary layer positions, turbulence shifts from rod-like (curve II) toward disk-like (curve III) states. Here, differences between uncontrolled and suction cases become less noticeable. For the blowing case (X087A150 in figure 13
d),
$y^+ \approx 30$
values have not been provided due to the presence of near-wall reverse flow. Here, disk-like turbulent structures appear near the wall, while rod-like structures emerge farther from the wall. In summary, blowing control induces distinct anisotropy patterns across the entire boundary layer, whereas suction control primarily affects the near-wall region. Among suction cases, the optimal X087A059 demonstrates the most significant alterations in the logarithmic layer.
Anisotropy invariant map with different wall-normal distances at
$x=0.96l_c$
. The two-component turbulence, rod-like axisymmetric turbulence and disk-like axisymmetric turbulence regimes are demarcated by black dashed lines I, II, and III, respectively. Here 1C, 2C, and 3C denote single-component turbulence, two-component axisymmetric turbulence and isotropic turbulence, respectively. The square symbols in the figure represent the location at
$y^+ \approx 30$
, while the upper portion of the curves corresponds to smaller
$y^+$
values.
Considering that the multi-process acoustic theory (Mao et al. Reference Mao, Kang, Wu, Yu, Gao, Su and Lu2020) offers a robust method for quantifying changes in noise source structures, figure 14 illustrates the spatial distribution of instantaneous shear acoustic sources near the trailing edge across various cases. In the uncontrolled case, the acoustic source structure is predominantly concentrated near the trailing edge. Suction significantly diminishes this structure, while the optimal case (X087A059) exhibits a markedly weakened acoustic source structure compared with case X093A059. Conversely, blowing (X087A150) substantially amplifies the acoustic source structure, extending it into the wake.
Isocontours of the instantaneous physical noise source structures for (a) the uncontrolled case, (b) the optimal case X087A059, (c) the suction case X093A059, and (d) the blowing case X087A150. The isosurfaces of
$-\beta = \pm 100$
(blue colour for
$-\beta = 100$
and red colour for
$-\beta = -100$
) are displayed near the trailing edges.
In summary, blowing control induces additional vortex shedding, significantly altering the flow structure and markedly increasing the acoustic source intensity, which drives the observed rise in low-frequency noise. Conversely, suction control tends to disrupt the flow structure near the wall, reducing the acoustic source intensity. The mechanisms behind this reduction and the noise suppression in the optimal suction control case (X087A059) require further investigation. The noise reduction mechanism for optimal control will be explored in the next subsection.
4.3. Noise reduction mechanisms
As evidenced in the above subsection, suction control modifies turbulent structures in the near field. These modifications to the flow structures inevitably affect the wall-pressure fluctuations near the trailing edge, leading to a reduction in far-field noise. Here, pressure–velocity coherence analysis is employed to understand the effect of turbulent motion on wall-pressure fluctuations (Pan & Ye Reference Pan and Ye2025). The pressure–velocity coherence definition proposed by Zang, Mayer & Azarpeyvand (Reference Zang, Mayer and Azarpeyvand2020) is utilised to eliminate spurious coherence regions, expressed as
\begin{align} C_{p^{\prime } u_{\kern-1pt i}^{\prime }}(St, y^+)=\frac {\big |\phi _{p^{\prime } u_{\kern-1pt i}^{\prime }}(St, y^+)\big |^2}{\phi _{p^{\prime } p^{\prime }}(St) \phi _{u_{\kern-1pt i}^{\prime } u_{\kern-1pt i}^{\prime }}(St, y^+)}\left (\frac {u^{\prime }_{\textit{rms}}(y^+)}{U_0}\right )^2 \!, \end{align}
where
$\phi _{p^{\prime } p^{\prime }}$
and
$\phi _{u_{\kern-1pt i}^{\prime } u_{\kern-1pt i}^{\prime }}$
are the power spectral density of the pressure and velocity fluctuations, respectively. Here
$\phi _{p^{\prime } u_{\kern-1pt i}^{\prime }}$
denotes the cross-spectral density between them;
$u_1^{\prime }$
and
$u_2^{\prime }$
represent the tangential and normal velocity fluctuations, respectively;
$u^{\prime }_{\textit{rms}}$
is the root mean square of the tangential velocity fluctuation. Figure 15 presents the contribution of tangential velocity fluctuations to wall-pressure fluctuations. As shown in figure 15(a), the uncontrolled case exhibits a high-coherence region in the buffer layer (
$5 \lesssim y^+ \lesssim 30$
) across the investigated frequency range (
$St$
= [0,20]). This indicates a strong correlation between the tangential component of flow structures in this region and wall-pressure fluctuations, consistent with the experimental findings of Szőke et al. (Reference Szőke, Fiscaletti and Azarpeyvand2020a
), which also report elevated coherence in the buffer layer. For cases X087A059 and X087A017, the location of the high-coherence region remains unchanged; however, both the extent and magnitude of coherence are reduced to varying degrees. In contrast, when suction is applied closer to the trailing edge, the coherence characteristics differ significantly. As shown in figure 15(d), a substantially enhanced high-coherence region emerges in the logarithmic region. This enhancement of low-frequency coherence may contribute to the reduced effectiveness of suction in case X093A059.
Pressure–velocity coherence
$C_{p^{\prime } u_1^{\prime }}$
at
$x/l_c = 0.96$
: (a) the uncontrolled case, (b) the optimal case X087A059, (c) the suction case X087A017, and (d) the suction case X093A059.
Unlike the distribution of
$C_{p^{\prime } u_1^{\prime }}$
, the contribution of normal fluctuations to wall-pressure fluctuations is primarily concentrated in the medium-to-high frequency range (see figure 16) (Pan & Ye Reference Pan and Ye2025). For the uncontrolled case, the values of
$C_{p^{\prime } u_2^{\prime }}$
are larger than those of
$C_{p^{\prime } u_1^{\prime }}$
across the entire spectrum. This implies that the squeezing or lifting of vortex structures exerts a more significant impact on wall-pressure fluctuations. Additionally, the peak location of
$C_{p^{\prime } u_2^{\prime }}$
also lies within the buffer layer. Suction control notably attenuates the contribution of normal vortex motion at the trailing edge to wall-pressure fluctuations, as depicted in figures 16(b) and 16(c). The coherence level in the optimal case X087A059 is relatively lower than that in X087A017. Undoubtedly, the attenuation of vortex-wall coherence facilitates noise reduction. Figure 16(d) reveals that while suction near the trailing edge attenuates the vortex-wall interactions within the buffer layer, it induces elevated coherence in both the viscous sublayer and logarithmic layer. Given the extensive spatial extent of the logarithmic region, the increased
$C_{p^{\prime } u_2^{\prime }}$
magnitudes at mid-frequencies might impede noise reduction efforts. Moreover, in most regions of figures 15(d) and 16(d), the magnitude of
$C_{p^{\prime } u_1^{\prime }}$
exceeds that of
$C_{p^{\prime } u_2^{\prime }}$
. This indicates that suction control in case X093A059 promotes the dominance of streamwise shear and convection of large-scale structures in governing wall-pressure fluctuations at the trailing edge.
Pressure–velocity coherence
$C_{p^{\prime } u_2^{\prime }}$
at
$x/l_c = 0.96$
: (a) the uncontrolled case, (b) the optimal case X087A059, (c) the suction case X087A017, and (d) the suction case X093A059.
To further elucidate why X087A059 achieves the optimal control effect, a quantitative comparison of the sound source intensity distributions near the trailing edge is conducted across different suction scenarios, as illustrated in figure 17. Suction markedly reduces the physical noise source intensity in the near field, with the optimal control case (X087A059) exhibiting the lowest
$\sqrt {\langle \beta ^2\rangle }$
values across all inner layers of the boundary layer. Notably, in the logarithmic region, X087A059 demonstrates a substantial reduction in physical sound source intensity compared with other suction cases. Additionally, the location of the peak sound source shifts. In the uncontrolled case, the maximum
$\sqrt {\langle \beta ^2\rangle }$
occurs at
$y^+ \approx 20$
, whereas for suction cases applied away from the trailing edge (X087A059 and X087A017), the peak shifts closer to the wall at
$y^+ \approx 10$
. It is noteworthy that the regions of high coherence in figures 15 and 16 coincide exactly with the buffer layer. However, when suction is applied near the trailing edge, the physical sound source intensity in the viscous sublayer is significantly amplified.
Profiles of the physical noise source intensity
$\sqrt {\langle \beta ^2\rangle }$
as a function of dimensionless wall distance at
$x/l_c = 0.96$
.
To gain a deeper understanding of the factors contributing to variations in sound source intensity, we utilise the trailing-edge scattering noise prediction model, known as the TNO model, developed by Parchen (Reference Parchen1998). The TNO model comprises two components: (i) the wavenumber–frequency wall-pressure spectrum derived by Blake (Reference Blake1986), and (ii) the prediction of far-field noise using trailing-edge diffraction theory (Howe Reference Howe1978). For observer positions directly above the trailing edge, the TNO model’s prediction can be expressed as (Kamruzzaman et al. Reference Kamruzzaman, Herrig, Lutz, Würz, Krämer and Wagner2011)
\begin{align} \varPhi _{pp}(k_1, k_3, \omega ) & = 4\rho ^2 \left ( \frac {k_1^2}{k_1^2 + k_3^2} \right ) \int _0^{\infty } \left [ \frac {\partial U_1(y_2)}{\partial y_2} \right ]^2 \varLambda _y(y_2) \big\langle u_2^{\prime 2}(y_2) \big\rangle \nonumber \\& \quad \boldsymbol{\cdot}\widetilde {\phi }_{22}(k_1, k_3, y_2) \phi _{m}(\omega - k_1 U_c) e^{-2|k|y_2} \mathrm{d} y_2, \\[-28pt] \nonumber \end{align}
\begin{align} \mathrm{S}_{pp}(\omega )=\frac {\mathrm{L}}{4 \pi \mathrm{R}^2} \int _{-\infty }^{\infty } \frac {\omega }{\mathrm{c}_0 k_1} \varPhi _{pp}\left (k_1, 0, \omega \right ) \mathrm{d}k_1, \\[2pt] \nonumber \end{align}
where
$\varPhi _{pp}$
denotes the wavenumber–frequency spectrum of wall-pressure fluctuations and
$\mathrm{S}_{pp}$
represents the far-field pressure power density spectrum. The variables
$k_1$
,
$k_3$
,
$\omega$
,
$U_1$
and
$y_2$
denote the streamwise wavenumber, spanwise wavenumber, angular frequency, tangential velocity and wall-normal position, respectively. Here
$k=\sqrt {k_1^2+k_3^2}$
. The parameters
$L$
and
$R$
denote the trailing-edge wetted length and the distance from the trailing edge to the observer, respectively. In the TNO model, the near-field sound source intensity is determined by the velocity gradient
$ {\partial U_1(y_2)}/{\partial y_2}$
, vertical integral length scale
$\varLambda _y(y_2)$
and mean square value of wall-normal velocity fluctuations
$\langle u_2^{\prime 2}(y_2) \rangle$
. Under the frozen turbulence hypothesis, the moving-axis spectrum
$\phi _{m}(\omega - k_1 U_c)$
can be reduced to a Dirac delta function. The normalised spectrum of vertical velocity fluctuations
$\widetilde {\phi }_{22}$
(normalised by
$\langle u_2^{\prime 2}(y_2) \rangle$
) determines the dominant source frequency, while the term
${\rm e}^{-2|k|y_2}$
tends to reduce the frequency of peak noise (Zhou et al. Reference Zhou, Zhong and Zhang2021). Thus, we can regard
$ ( {\partial U_1(y_2)}/{\partial y_2} )^2 \varLambda _y(y_2) \langle u_2^{\prime 2}(y_2) \rangle$
as a nominal source term in the TNO model (Wolf et al. Reference Wolf, Lutz, Würz, Krämer, Stalnov and Seifert2015). Below, we analyse the effects of suction control on each component individually. Additionally, in some previous studies (Kamruzzaman et al. Reference Kamruzzaman, Lutz, Herrig and Krämer2012), the calculation of the wall-pressure spectrum relied on isotropic modelling, which required the inclusion of an anisotropic correction factor (such as
$f_{22}^{aniso} = {\langle u_2^{\prime 2} \rangle }/{2/3k_T}$
, where
$k_T$
represents the turbulent kinetic energy). However, in the current research, the TNO source term is obtained from large eddy simulations, so anisotropic correction is not considered. When evaluating far-field noise, the spectrum of wall-pressure fluctuations is typically measured near the trailing edge but at a sufficient distance to capture the incident pressure field of boundary layer turbulence (Sanjosé et al. Reference Sanjosé, Moreau, Lyu and Ayton2019). Accounting for the suction slot position, we examine turbulence information at
$x/l_c = 0.96$
.
Figure 18(a) illustrates the modifications to the turbulent boundary layer velocity profile at the trailing edge induced by suction control. For suction control away from the trailing edge (X087A059 and X087A017), the influence on the viscous sublayer (
$y^+ \lesssim 5$
) is diminished, while the dimensionless velocity in the buffer layer (
$5 \lesssim y^+ \lesssim 30$
) exhibits a slight increase. In contrast, the effect of suction on the logarithmic region is most pronounced, with the velocity profile shifting towards the log-law. For the suction slot at
$x/l_c \approx 0.87$
, suction control with an angle of
$\alpha \approx 0.59\pi$
induces more substantial modifications to the logarithmic region. Due to the closer proximity of the trailing edge to the suction slot, the boundary layer changes in X093A059 are more distinct, particularly in regions of
$y^+\gt 10$
, where the dimensionless velocity shows a large decrease. However, in the TNO model, the velocity gradient plays an important role. Notably, while X093A059 exhibits the most significant velocity profile changes, its reduction in velocity gradient is less effective than that in X087A059. Figure 18(b) further presents the variation in velocity gradient within the logarithmic region induced by suction. Among all suction cases, the optimal case X087A059 achieves the greatest reduction in velocity gradient within the region where
$y^+\gt 30$
, while case X093A059 demonstrates greater efficacy in the buffer layer. Given the considerable extent of the log-law region, modifications to the velocity gradient in this zone contribute substantially to variations in the overall acoustic source strength.
The turbulent boundary layer profiles at
$x/l_c = 0.96$
: (a) the dimensionless velocity profiles, and (b) the dimensionless velocity gradient profiles (non-dimensionalised by the speed of sound and aerofoil chord length).
Figure 19 presents the distribution of turbulent structure length scales at the trailing edge for different cases. The vertical integral length scale is calculated according to the Wilcox model (Wilcox Reference Wilcox1998), namely,
\begin{align} \varLambda _y=\frac {1}{0.41} \sqrt {\frac {\left |\langle u_1^{\prime } u_2^{\prime } \rangle \right |}{\left (\partial U_1 / \partial y_2\right )^2}}, \end{align}
where
$u_1^{\prime }$
and
$u_2^{\prime }$
denote the velocity fluctuations tangential and normal to the wall, respectively. The spanwise length scale
$\varLambda _z$
is obtained directly by integrating the autocorrelation function of the spanwise velocity fluctuations. As can be seen in figure 19(a), the vertical integral length scales generally increase for both case X087A059 and case X087A017, except for a few specific locations. In contrast, case X093A059 exhibits increased vertical integral length scales only in the near-wall region, with significant reductions observed where
$y_2/\delta _0 \gt 0.2$
. The behaviour of the spanwise integrated length scales, however, exhibits distinct behaviour, as illustrated in figure 19(b). In the near-wall region,
$\varLambda _z$
exhibits an increase, whereas in regions farther from the wall (
$y_2/\delta _0 \gt 0.2$
), these scales decrease to varying degrees across all suction cases, with case X087A059 showing the most pronounced reduction. In the TNO model, only the vertical integral length scales influence wall-pressure fluctuations. The results in figure 19(a) suggest that variations in length scales are not the primary driver of noise reduction, a finding consistent with the experiments by Wolf et al. (Reference Wolf, Lutz, Würz, Krämer, Stalnov and Seifert2015).
Length scales of turbulent structures at
$x/l_c = 0.96$
: (a) the vertical integral length scale
$\varLambda _y$
, and (b) the spanwise integrated length scale
$\varLambda _z$
, where
$\delta _0$
represents the boundary layer thickness at the trailing edge for the uncontrolled case.
Within the nominal acoustic source term of the TNO model, the wall-normal Reynolds stress component also plays a critical role. As shown in figure 20, suction induces varying degrees of reduction in the wall-normal Reynolds stress component across different boundary layer regions. Notably, substantial attenuation occurs within the log-law region. Furthermore, case X087A059 exhibits the weakest wall-normal velocity fluctuations throughout the boundary layer, a key factor contributing to its maximum noise reduction. Additionally, certain suction controls shift the peak location of the wall-normal Reynolds stress component. In the uncontrolled case, the peak resides at
$y^+ \approx 40$
. For cases X087A059 and X087A017, the peak location shifts away from the wall to
$y^+ \approx 50$
; while case X093A059 shows negligible displacement, maintaining its peak position.
Profiles of the Reynolds stress component in the wall-normal direction
$\langle u_2^{\prime 2} \rangle$
as a function of dimensionless wall distance at
$x/l_c = 0.96$
.
The total nominal source term of the TNO model
$ ( {\partial U_1}/{\partial y_2} )^2 \varLambda _y \langle u_2^{\prime 2} \rangle$
is examined in figure 21. In the uncontrolled case, the peak of the TNO source term resides at
$y^+ \approx 20$
, closely aligned with the peak location of the physical acoustic source strength
$\sqrt {\langle \beta ^2\rangle }$
shown in figure 17. In suction-controlled cases the magnitude of the TNO source term is markedly reduced. The most substantial attenuation occurs in the logarithmic region, particularly in the optimal noise reduction case (X087A059). Given the extensive spatial extent of the logarithmic region, the source term in this region significantly contributes to the overall radiated noise. Additionally, the source term in suction cases exhibits a dual-peak structure. The peak locations for cases X087A059 and X087A017 are closely aligned but shifted farther from the wall compared with case X093A059. The generation of the dual-peak structure results from the combined effects of changes in boundary layer velocity gradients, wall-normal Reynolds stresses and variations in the vertical integral length scale. This suggests that suction control may have altered the primary region of noise generation – shifting it from being dominated by the buffer layer to a joint dominance by both the buffer layer and the logarithmic layer. A comparison of figures 17 and 21 reveals that, despite differences in the distributions of the physical noise source and the TNO source term in suction cases, both consistently highlight the critical role of source attenuation in the logarithmic region for achieving optimal noise reduction in case X087A059. From an engineering perspective, a suction control device placed at a distance from the trailing edge and slightly angled towards the upstream direction can more effectively reduce the source strength and lower the far-field noise.
Profiles of
$( {\partial U_1}/{\partial y_2} )^2 \varLambda _y \langle u_2^{\prime 2} \rangle$
as a function of dimensionless wall distance at
$x/l_c = 0.96$
.
In short, case X087A059 dramatically suppresses pressure–velocity coherence within the boundary layer, thereby facilitating noise reduction, whereas the suction near the trailing edge (X093A059) amplifies the footprint of logarithmic layer turbulence in wall-pressure fluctuations. Moreover, case X087A059 achieves optimal noise reduction primarily by substantially diminishing the velocity gradient at the trailing edge and suppressing wall-normal Reynolds stress components, thereby inducing a marked decrease in sound source intensity within the logarithmic region. The alterations in turbulence length scales do not play decisive roles in the noise attenuation mechanism.
5. Summary and conclusions
In this study, ILES are conducted on a NACA0012 aerofoil with and without trailing-edge blowing/suction. The results are obtained under constant flow control intensity (
$C_\mu = 0.8$
) and fixed control slot length (
$b = 0.05l_c$
). The centre position and angle of the active control region are described as optimisation variables, with respective value ranges of
$[0.825l_c, 0.975l_c]$
and
$[0, 2\pi ]$
. To achieve maximum noise reduction, two rounds of optimisation are applied: direct Bayesian optimisation and sound-source-oriented coarse-grained optimisation. Some typical cases are examined to elucidate changes in far-field noise and near-field turbulence structures, while the underlying noise reduction mechanisms under optimal control settings are investigated.
In the direct Bayesian optimisation process, the mean overall SPL (
$\overline {\Delta \text{OASPL}}$
) in the far field is selected as the objective function, while the parallel upper confidence bound (q-UCB) policy is employed as the acquisition function. Each optimisation iteration requires the execution of five high-fidelity 3D simulations. For the first round of Bayesian optimisation, 15 iterations are conducted, with the optimal noise reduction parameters obtained at the fifth iteration. The final optimal parameters include a suction location at
$x/l_c=0.8734$
and an angle of
$\alpha /\pi = 0.5895$
, resulting in a reduction of
$\overline {\Delta \text{OASPL}}$
by up to 3.7 dB.
To reduce the computational cost of direct Bayesian optimisation, the surrogate model is constructed based on the physical noise source defined by the multi-process acoustic theory. This approach works because statistically processing physical sound sources offers low signal latency and quick convergence, which dramatically reduces the cost of the optimisation workflow. It should emphasise that the GP models constructed by noise reduction and physical source intensity demonstrate similar landscapes. Specifically, all blowing control amplifies both sound source intensity and noise levels. Suction adjacent to the trailing edge exhibits poor noise reduction efficacy. As the suction slot shifts upstream, effective control angles expand progressively from wall-normal suction toward broader orientations. The optimal control parameters obtained by both GP models are similar. The GP model demonstrates the feasibility of sound-source-oriented coarse-grained Bayesian optimisation, even though its physical source cannot capture multiple extrema in the noise reduction landscape. Subsequently, we implement this source-oriented Bayesian optimisation, achieving comparable noise reduction within merely two iterations. Under the current statistical strategy, the source-oriented Bayesian optimisation reduces the cost per sampling by
$60\,\%$
. Considering the reduction in required iterations, the source-oriented optimisation can reduce the total computational cost to approximately one-sixth of the initial direct optimisation.
Next, the changes in noise and flow structures under blowing or suction control are examined through several typical cases. The blowing control generates additional large-scale vortex shedding, which significantly enhances noise radiation within the low-frequency range (
$St = [2,6]$
). In the most energetic DMD mode of the blowing case (X087A150), an antisymmetric structure associated with this vortex shedding is observed. Furthermore, the suction control does not shift the dominant frequency band of the DMD modes (
$St = [5,6]$
), but it substantially reduces the mode energy across all frequencies. Among these suction cases, X087A059 demonstrates the most pronounced effect, effectively disrupting the coherent mode structures in the near field and thereby achieving the greatest reduction in both mode energy and radiated noise. The anisotropic invariant map indicates that the suction control has a significant impact on the near-wall turbulent structures, with case X087A059 exhibiting the most substantial alterations in the logarithmic layer.
Finally, pressure–velocity coherence analysis reveals that the optimal suction control (X087A059) substantially reduces pressure–velocity coherence within the boundary layer, contributing to effective noise reduction. In contrast, suction near the trailing edge (X093A059) significantly amplifies the footprint of logarithmic layer turbulent motions within surface pressure fluctuations, potentially explaining its suboptimal acoustic performance. Quantitative analysis of physical noise sources reveals that case X087A059 exhibits the weakest sound source intensity in the logarithmic layer, which can be regarded as the primary factor contributing to its optimal noise reduction performance. Using the TNO model, it is found that the decrease in logarithmic layer velocity gradients and diminishing wall-normal Reynolds stresses dominate the attenuation of noise source strength, while changes in turbulent length scales have minimal impact.
In reality, this work introduces a novel framework that integrates Bayesian optimisation with advanced noise diagnostic theory. This framework effectively achieved active control to mitigate TBL-TE noise at a significantly reduced cost. The elucidated noise reduction mechanism offers valuable insights for future TBL-TE noise control strategies. Furthermore, this study suggests that a source-informed, multi-fidelity Bayesian optimisation framework could also be utilised to address multi-objective design challenges in aerodynamics and aeroacoustics in the future.
Acknowledgements
The numerical computations in this paper have been performed at Hefei Advanced Computing Center.
Funding
This work is supported by the National Natural Science Foundation of China under grant Nos. 12388101, 12322210, 12402317, 92252202, and 12172351, the Strategic Priority Research Program of the Chinese Academy of Sciences (Grant No. XDB0500301) and the Fundamental Research Funds for the Central Universities.
Declaration of interests
The authors report no conflict of interest.
Data availability statement
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Appendix. Selection of the sound source integration domain
To illustrate the impact of integration volume selection on
$\beta ^*$
, we compared the GP model from figure 6 with another model based on a smaller integration volume. It can be observed that altering the integration domain only slightly affects the predicted distribution of
$\beta ^*$
(see figure 22). The noise reduction trends and the identified optimal parameter regions obtained from different integration domains are generally consistent. Additionally, it should be noted that the objective of the present optimisation framework is to indicate the region where the optimal control parameters are located. This can be achieved as long as our integration volume covers the primary variation region of the sound source. For more precise noise reduction capture, a multi-fidelity Bayesian approach should be employed.
The GP models based on near-field physical noise source strength
$-\beta ^*$
: (a) the integration volume
$V\in [l_c-2\delta _0,l_c+\delta _0] \times [-1.5\delta _0,1.5\delta _0]\times [0,0.1l_c]$
; (b) the integration volume
$V\in [l_c-\delta _0,l_c+ 0.5\delta _0] \times [-\delta _0,\delta _0]\times [0,0.1l_c]$
.
