Extracting the Spatial Correlation of Wall Pressure Fluctuations Using Physically Driven Artificial Neural Network

Abstract

The spatial correlation of wall pressure fluctuations is a crucial parameter that affects the structural vibration caused by a turbulent boundary layer (TBL). Although the phase-array technique is commonly used in industry applications to obtain this correlation, it has proven to be effective only for moderate frequencies. In this study, an artificial neural network (ANN) method was developed to calculate the convective speed, indicating the spatial correlation of wall pressure fluctuations and extending the frequency range of the conventional phase-array technique. The developed ANN system, based on a radial basis function (RBF), has been trained using discrete simulated data that follow the physical essence of wall pressure fluctuations. Moreover, a normalization method and a multi-parameter average (MPA) method have been employed to improve the training of the ANN system. The results of the investigation demonstrate that the MPA method can expand the frequency range of the ANN, enabling the maximum analysis frequency of convective velocity for aircraft wall pressure fluctuations to reach over 10 kHz. Furthermore, the results reveal that the ANN technique is not always effective and can only accurately calculate the wavenumber when the standard wavelength is less than four times the width of the sensor array along the flow direction.

1. Introduction

Wall pressure fluctuations generated by a turbulent boundary layer (TBL) are of significant concern in aerospace engineering applications. In the case of aircraft, these fluctuations are the primary source of structural vibrations [1,2]. Wall pressure fluctuations transmit in the form of acoustic energy into the equipment compartment and can also affect the reliability of sound-sensitive equipment in the compartment. Understanding the mechanisms and laws governing wall pressure fluctuations is crucial for mitigating structural vibrations and reducing background noise in submarines caused by TBLs. Furthermore, accurate excitation models are essential for enhancing the design of different vehicle types, in addition to providing a deeper understanding of flow dynamics.

To investigate wall pressure fluctuations beneath a turbulent boundary layer (TBL), the commonly employed approaches include numerical computations [3,4,5] and measurements [2,6,7]. Computational Fluid Dynamics (CFD) is an effective numerical approach for calculating flow details and understanding vortex evolution mechanisms. Kokkinakis [8] conducted Direct Numerical Simulations (DNSs) to study shock wave boundary layer interactions, revealing a low-frequency λ-shock oscillation caused by the interaction between the λ-shock and the post-shock relaxation region. Inoue [9] utilized a Large Eddy Simulation (LES) to investigate TBLs with zero-pressure gradient on a smooth flat plate, analyzing the effect of Reynolds number on flow structures. CFD has proven effective in establishing the relationship between wall pressure fluctuations and vortex structures, while wind tunnel tests accurately evaluate the intensity of these fluctuations under different TBL conditions, providing validation for numerical computations. Laurence [10] conducted visualization experiments in a reflected-shock wind tunnel to illustrate instability wave development within the boundary layer on a slender cone at high Mach numbers, highlighting energy concentration toward the wall under a variable wall-normal distribution in low-enthalpy scenarios.

Previous research has considered the intensity and power spectral density of wall pressure fluctuations; however, these measures have proven inadequate in explaining the mechanism behind structural vibrations caused by these fluctuations [11]. Zhao [11] concluded that the spatial correlation of wall pressure fluctuations also plays a crucial role in both initiating and suppressing structural vibrations. To describe this spatial correlation, various wavenumber-frequency (W-F) spectra models have been proposed, such as the Corcos model [12], Chase model [13], and Efimtsov model [14]. With an improved understanding of wall pressure fluctuations, these classic models have undergone modifications and enhancements. Doisy [15] derived a wave vector-frequency spectrum based on Lighthill’s equation for wall pressure fluctuations beneath a rigid flat plate at low Mach numbers. Caiazzo [16] introduced the generalized Corcos model, offering better control of decay in the wavenumber domain below the convective peak while preserving important mathematical properties. Nevertheless, existing W-F models only provide approximate ranges of wavenumbers for different wall pressure fluctuation modes, which are insufficient for investigating the spatial correlation of structural vibrations.

To accurately determine the spatial correlation of wall pressure fluctuations, wind tunnel tests have proven effective in measuring the wavenumber of wall pressure fluctuations on the surface of aircraft. Hu [17] utilized an L-shaped pressure sensor array to measure the coherence of wall pressure fluctuations in zero and adverse pressure gradients. Zhao [18,19] introduced an improved technique using a phased array to identify the wavenumbers of wall pressure fluctuations and applied it to obtain the spatial correlation under hypersonic flat-plate flow. While measurements can directly capture the spatial correlation of wall pressure fluctuations, they are limited to moderate-frequency bands when it comes to wavenumber measurements, as indicated by Zhao [20]. Errors in obtaining low-frequency and high-frequency wavenumbers mainly arise from the length and density of the sensor array [20]. To expand the frequency range of spatial correlation determination using the phase-array technique for wall pressure fluctuation measurements, an artificial neural network (ANN) is designed to extract the spatial correlation from the measured wall pressure fluctuations based on physically driven principles. In fact, ANNs have demonstrated excellent performance in solving multivariable and nonlinear problems through classification [21,22] and regression [23,24] in the field of aerodynamics [25,26] over the past decade. The present work aims to address the limitations of the conventional phase-array approach by incorporating an ANN.

The structure of this paper is as follows: Section 2 provides an introduction to the fundamental theory of the investigation. Section 3 and Section 4 discuss the validation and application of the ANN method, respectively. The analysis of errors is presented in Section 4, and the conclusions are given in the final section.

2. Basis Theory

2.1. Wall Pressure Fluctuation Model

W-F models, such as the Corcos model [12], Chase model [13], and Efimtsov model [14], are commonly used to describe wall pressure fluctuations. These models often have different expressions to suit various flow scenarios. However, the spatial correlation of wall pressure fluctuations is solely dependent on the phase change along the airflow direction, which allows for the simplification of all W-F models into a uniform expression [18]:

$$S_{pp}\left({\xi }_x,{\xi }_y,\omega \right)=S_p\left(X,\omega \right)e^{-j{\displaystyle \frac{\omega {\xi }_x}{U_c}}},$$

(1)

where $U_{\mathrm{c}}$ represents the convective velocity, which determines the spatial correlation of wall pressure fluctuations and was the main focus of investigation in this study. ${\xi }_x$ and ${\xi }_y$ denote the distances between any two points on the model surface in the flow and span directions, respectively. $\omega$ is the angular frequency, and $S_{pp}$ refers to the cross-spectral density of the wall pressure fluctuations. $j\omega {\xi }_x/U_{\mathrm{c}}$represents the phase difference in the wall pressure fluctuations between two arbitrary points along the flow. X represents the relative coordinates of the two arbitrary points on the model surface, and $S_p\left(X,\omega \right)$ represents the cross-spectral density function of the wall pressure fluctuations. The expressions for $S_p\left(X,\omega \right)$ in describing the power spectrum distribution of the wall pressure fluctuations differ for different W-F models.

Equation (1) illustrates that the cross-spectral density of wall pressure fluctuations at two distinct locations on the test model can be represented as the correlation between the pressures at two arbitrary points [18]. The wall pressure fluctuations caused by a turbulent boundary layer (TBL) consist primarily of two components [27]: the acoustic mode and the hydrodynamics mode. As depicted in Figure 1, $k_x$ and $k_y$ signify the wavenumbers in the flow direction and span direction, respectively, while $\Phi \left(k,\omega \right)$ represents the amplitude of the W-F model. The hydrodynamics mode is a non-radiating element associated with the speed of convective airflow, which can be obtained from Equation (1) as:

$$p_{TBL}\left({\xi }_x,{\xi }_y,\omega \right)=p_{TBL}\left(X,\omega \right)e^{-j{\displaystyle \frac{\omega {\xi }_x}{U_c}}},$$

(2)

where $p_{TBL}\left(X,\omega \right)$ denotes the magnitude of wall pressure fluctuations induced by the turbulent boundary layer (TBL) on the model structure. Furthermore, the convective velocity, which governs the spatial correlation of the pressure fluctuation, can be expressed as:

$$k_c={\displaystyle \frac{\omega }{U_c}}.$$

(3)

Clearly, the spatial correlation of wall pressure fluctuations is determined by the convective speed $U_c$, as depicted in Figure 1. Classical W-F models suggest that the convective speed beneath a turbulent boundary layer (TBL) should fall within the range of $0.5U_0\le U_c\le 0.7U_0$ in the entire frequency spectrum [28], where $U_0$ represents the freestream velocity. However, this wide range of convective speeds fails to meet the requirements for precise design of structural vibration suppression.

The acoustic mode is a radiating component that propagates with the speed of sound. It can be mathematically represented by the wave equation as:

$$p_a\left({\xi }_x,\omega \right)=p_a\left(\omega \right)e^{-j{\displaystyle \frac{\omega {\xi }_x}{c_0+U_s}}},$$

(4)

where $p_a\left(\omega \right)$ represents the amplitude of aerodynamic noise sound pressure, $\omega$ is the angular frequency, $c_0$ is the speed of sound, $U_s$ is the local flow velocity, and ${\xi }_x$ denotes the variation in distance along the streamwise direction. Figure 1 displays the propagation directions of acoustic wavenumbers $k_0$, which can be determined as follows:

$$k_0={\displaystyle \frac{\omega }{c_0}}$$

(5)

Considering that the coordinates of the sensors on the test model undergo minimal changes, it is assumed that the amplitude of the acoustic mode does not attenuate in the flow direction or around the test model. By referring to Equations (2) and (4), the total wall pressure fluctuations on the model surface can be expressed as:

$$p=p_{TBL}\left({\xi }_x,{\xi }_y,\omega \right)+p_a\left({\xi }_x,\omega \right).$$

(6)

Significantly, the key distinction between the hydrodynamics mode and the acoustic mode lies in their spatial correlations, which are determined by $U_c$ and $c_0+U_s$, respectively, as demonstrated in Equations (2) and (4).

2.2. Conventional Method

To determine the spatial correlation of wall pressure fluctuations, the conventional phase-array technique is commonly utilized, relying on measurement data [18]. This technique employs a beamforming process to calculate the wavenumber, which represents the spatial correlation of wall pressure fluctuations [18,19]. The calculation can be expressed as:

$$b\left(k_x,k_y,\omega \right)=e^T\left(k_x,k_y\right)D\left(\omega \right)e\left(k_x,k_y\right),$$

(7)

where $k_x$ and $k_y$ represent the wavenumbers in the flow direction and span direction, respectively. The cross-spectrum matrix $D\left(\omega \right)$ is composed of discrete cross $S_{pp}\left({\xi }_x,{\xi }_y,\omega \right)$, and $e_m\left(k_x,k_y\right)$ signifies the steering vector.

$$e_m\left(k_x,k_y\right)=\exp \left[i\left\{x_m{k}_x+y_m{k}_y\right\}\right]$$

(8)

To simplify the expression and calculation, Equation (7) can be simplified to the following equation as

$$b\left(k_x,k_y,\omega \right)={\alpha }^T\left(k_x,k_y\right)S\left(\omega \right),$$

(9)

where

$${\alpha }_j\left(k_x,k_y\right)={\left\{e_m\left(k_x,k_y\right)e_n^{*}\left(k_x,k_y\right)\right\}}_j=\exp \left[i\left\{k_x{\xi }_j+k_y{\eta }_j\right\}\right].$$

The discrete separation between two sensors can be described by ${\xi }_j={\left\{x_m-x_n\right\}}_j$ and ${\eta }_j={\left\{y_m-y_n\right\}}_j$, where $S\left(\omega \right)$ represents a vector containing the elements of the matrix $D\left(\omega \right)$. Furthermore, $N$ denotes the total number of sensors utilized for the measurement, with $m$ and $n$ ranging from 1 to $N$, and j ranging from 1 to $N^2$. The wavenumber of wall pressure fluctuations can be determined by the beam map calculated through the beamforming process. Meanwhile, the phase-array technique, as depicted in Equation (7), can be employed to accurately obtain the spatial correlation based on the measured wall pressure fluctuations.

2.3. Physically Driven ANN

The use of radial basis functions (RBFs) in Artificial Neural Networks (ANNs) for this study is motivated by their simplicity and fast convergence in data processing, mathematical modeling, and reliability analysis [29,30]. Typically, an RBF network consists of three forward network layers, as depicted in Figure 2. The first layer is the input layer, with nodes equal to the input dimension. The second layer is the hidden layer, with the number of nodes determined by the complexity of the problem. The third layer is the single output layer. In cases where there are multiple outputs, the problem can be analyzed as a combination of multiple subproblems [31]. In recent years, RBF neural networks have been widely applied in various industrial fields due to the advancements of ANN in data analysis.

For this study, the ANN is employed to calculate the convective speed, which determines the spatial correlation of wall pressure fluctuations. To train the RBF network, the simulated distribution of wall pressure fluctuations, including the phase and amplitude, is provided by Equation (6) based on the understanding of the physical properties of wall pressure fluctuations. The measurement data alone does not contain sufficient information to obtain the spatial correlation of both high-frequency and low-frequency components [20]. However, this limitation can be compensated by the traveling wave equation (Equation (6)), which describes the distribution of wall pressure fluctuations. By training the ANN with the traveling wave equation that represents the physical law of wall pressure fluctuations, the RBF network can encompass the spatial correlation for the entire frequency range in theory. Since the spatial correlation of wall pressure fluctuations is determined by the convective speed, the focus of this study is on calculating the convective speed using the ANN. The main procedure for this calculation can be divided into several steps as follows:

Step I: Generation of Training Samples. In order to obtain the spatial correlation of wall pressure fluctuations, the convective velocity $U_c$ is chosen as the target value based on Equation (5). Furthermore, to mitigate the impact of varying flow velocities on the output value, the normalized ratio $r$ is employed as the training output value for the ANN:

$$r={\displaystyle \frac{U_c}{U_0}}$$

(10)

The matrix R, representing the output of the training process for the ANN, can be expressed as:

$$\mathbf{R}=\left[r_1,r_2,\cdots ,r_m\right]$$

(11)

where $r_1$ and $r_m$ represent the minimum and maximum values of the target value, respectively, and $R$ denotes a linearly increasing sequence, ranging from $r_1$ to $r_m$. It is important to note that various methods can be selected to expand the range of training samples based on the variation range of the target value, such as exponential growth, logarithmic growth, and power growth.

Corresponding to the output target value r, the simulated wall pressure fluctuations are computed as the training input samples for the RBF neural network. This can be expressed as:

$$P={\left[p_1,p_2,\cdots ,p_n\right]}^T,$$

(12)

where $p_i$ represents the pressure signal measured at the ith test point, which can be computed using the traveling wave equation, Equation (6). The node number of the input layer, denoted as n, is determined by the total number of sensors in the sensor array. It is important to note that although Equation (12) provides only discrete data, the physical mechanism of wall pressure fluctuation is incorporated in the expression of Equation (6).

The matrix $\mathbf{P}\mathbf{P}$, representing the input for the ANN training process, can be expressed as:

$$\mathbf{P}\mathbf{P}=\left[P_1,P_2,\cdots {,P}_m\right],$$

(13)

where

$$\left\{\begin{array}{c}{P}_1={\left[p_{1-1},p_{1-2},\cdots ,p_{1-n}\right]}^T\\ P_2={\left[p_{2-1},p_{2-2},\cdots ,p_{2-n}\right]}^T\\ ⋮\\ P_m={\left[p_{m-1},p_{m-2},\cdots ,p_{m-n}\right]}^T\end{array}.\right.$$

and $P_j$ represents the spatial distribution of the simulated wall pressure fluctuations measured by the sensor array under the corresponding condition of $r_j$. As illustrated in Figure 1, an RBF neural network is constructed with matrix PP as the input and matrix R as the output.

Step II: Data Preprocessing. As depicted in Figure 2, the wall pressure fluctuations obtained from the wind tunnel test using the sensor array can be represented as:

$$P_{sensor}={\left[p_{sensor-1},p_{sensor-2},\cdots ,p_{sensor-n}\right]}^T.$$

(14)

In order to calculate the spatial correlation of wall pressure fluctuations, the data need to be analyzed in the frequency domain. The signal $P_{sensor}$ collected by the sensor array in the time domain should, thus, be converted into the frequency domain using the fast Fourier transform (FFT), resulting in:

$$P_{FFT}={\left[p_{FFT-1},p_{FFT-2},\cdots ,p_{FFT-n}\right]}^T.$$

(15)

In general, the measured wall pressure fluctuations exhibit a wide variation range, which significantly impacts the accuracy of spatial correlation calculations by the RBF network. The amplitude of pressure fluctuations obtained from measurements is often much smaller compared to simulation data, which affects the accuracy of network calculations for spatial correlation. In the process of ANN debugging, it is a common practice to normalize the original measurement data to enhance the output accuracy of key parameters [22].

The expression for the pressure fluctuation signal obtained by the sensor array is given in Equation (12). The normalization formula is:

$$p_{norm-i}={\displaystyle \frac{p_{FFT-i}-\mathit{min}\left(P\right)}{\mathit{max}\left(P\right)-\mathit{min}\left(P\right)}},$$

(16)

where $1\le i \le n$, and $\mathit{max}\left(\mathbf{P}\right)$ and $\mathit{min}\left(\mathbf{P}\right)$ represent the maximum and minimum values of the matrix P, respectively. The normalized matrix can be expressed as:

$$P_{norm}={\left[p_{norm-1},p_{norm-2},\cdots ,p_{norm-n}\right]}^T.$$

(17)

Step III: Calculation Using RBF Networks. The type of hidden radial basis function is a key parameter that affects the performance of RBF networks, and should be determined first. In the hidden layer, each neuron is characterized by a Gaussian function, which can be described as:

$${\phi }_s\left(\mathbf{P}\right)=\mathit{exp}\left(-{\displaystyle \frac{{\sum }_{i=1}^n{\left(p_i-u_s\right)}^2}{2{{\sigma }_s}^2}}\right),$$

(18)

where $p_i$ represents the ith input element, and n denotes the number of input elements. ${\sigma }_s$ corresponds to the activation function of the RBF unit, while $u_s$ represents the mean value of the sth hidden unit, and ${\sigma }_s$ represents the covariance of the sth hidden unit. The parameters $u_s$ and ${\sigma }_s$ define the center and width of the Gaussian function, respectively, which are crucial characteristics for describing the RBF unit [31].

In this work, the output layer consists of a single output, which can be obtained as follows:

$$r={\sum }_{s=1}^q{\phi }_s{\epsilon }_s+{\epsilon }_{0,}$$

(19)

where ${\phi }_s$ represents the sth output of the hidden layer in Figure 1, ${\epsilon }_s$denotes the weight between the sth hidden unit and the output, ${\epsilon }_0$ signifies the bias weight of the output, and q represents the number of hidden units.

The convergence of the network is determined by the stop criterion, which is set to the mean square error ${{\rm E}}_{mse}$. The mean square error of the training samples is defined as follows:

$${{\rm E}}_{mse}={\displaystyle \frac{1}{m}}{\sum }_{j=1}^m{\left({\widehat{r}}_j-r_j\right)}^2,$$

(20)

where m represents the number of training samples, ${\widehat{r}}_j$ denotes the estimated output vector of the network, and $r_j$ represents the known target vector of the training samples. The choice of ${{\rm E}}_{mse}$ value will yield different training effects. In order to ensure sufficient training of the RBF networks, the stability of the network is tested using leave-one-out cross-validation. This is a common method where one item from the training sample is used as the test set, while the rest of the items serve as the training set. The error between the output of the test set and the target value is depicted in Figure 3. The red dots represent the target value, the blue dots represent the test output value, and the black line represents the test error.

As observed in Figure 3a, when ${{\rm E}}_{mse}$ is small, the network training is insufficient, resulting in the test output value being close to the mean of the preset range. Conversely, in Figure 3d, when ${{\rm E}}_{mse}$ is large, the network is overtrained, causing the test output value to exceed the preset range. Based on the results of the leave-one-out cross-validation, ${{\rm E}}_{mse}$ is set to a value smaller than ${10}^{-3}$. Using the output r from the RBF neural network, as shown in Figure 2, the convective velocity of the wall pressure fluctuation can be calculated using Equations (3) and (10). Subsequently, the spatial distribution of the hydrodynamics mode of wall pressure fluctuation can be calculated using Equation (2). The data density for the training input samples and output samples in Step I can be adjusted based on the accuracy of the separation results between the acoustic mode and hydrodynamics mode, until the mean square error requirements are satisfied.

3. Theoretical Validation

In order to assess the accuracy of the RBF neural network in determining the convective speed of wall pressure fluctuations, the results obtained through the ANN are compared with those obtained using the subsection approaching method (SAM) proposed by Zhao [32]. For the purpose of verification, the wall pressure fluctuations caused by a turbulent boundary layer (TBL) are measured using a linear array with 20 sensors and 0.05 m spacing [32]. To ensure the reliability of the ANN in identifying the wavenumber, the normalized convective speed is set between 0.1 and 1.0, and the desired training output can be defined as:

$$R=\left[r_1,r_2,\cdots ,r_n\right].$$

(21)

It is assumed that the amplitude of the hydrodynamics mode is at the same level as that of the acoustic mode, with no attenuation considered ($p_{TBL}\left(X,\omega \right)=p_a\left(\omega \right)=1Pa$) [32]. The total pressure measured at each sensor point is calculated using Equation (4). In the calculation, the speed of sound ($c_0$) is set to 340 m/s, while the freestream velocity ($U_0$) is set to 225 m/s. Based on the simulated wall pressure fluctuation, the ANN establishes a nonlinear relationship between the input signal and the desired output through RBF network training. Subsequently, the integrated pressure fluctuations at the target frequency are inputted into the trained ANN system to obtain the hydrodynamics mode of wall pressure fluctuation.

Based on the physical principles governing low-speed flow and the training samples used for the ANN, if the normalized convective velocity falls outside the range of 0.1 to 1.0, it indicates that the training error of the ANN is too large and cannot be referenced. To evaluate the performance of the ANN technique employed in this study, the lower limit of the preset identification frequency (LLPIF) is defined. This evaluation criterion ensures that the calculated convective velocity by the ANN falls within the preset range of the training samples (0.1–1.0). Generally, a smaller LLPIF corresponds to a broader frequency range for the ANN. It is important to note that although the normalized convective velocity in the frequency range higher than LLPIF may not fall within the reference range (0.5–0.7), it can still fall within the preset range (0.1–1.0).

The results of modal separation achieved by the ANN developed in this study, and the SAM proposed by Zhao [32] are presented in Figure 4 and Figure 5. The red dashed–dotted line represents the distribution of integrated wall pressure fluctuations along the flow direction, which can be calculated using Equation (6). The green solid line corresponds to the theoretical hydrodynamics mode given by Equation (2). The black dots and blue diamonds depict the calculated hydrodynamics modes obtained through the SAM [32] and the ANN developed in this study, respectively.

Figure 4 demonstrates that the hydrodynamics mode calculated by the ANN agrees well with that obtained through the SAM [32] and the theoretical results derived from Equation (2). The ANN effectively separates the hydrodynamics mode from the integrated wall pressure fluctuation in the moderate-frequency range. However, modal separation for low-frequency and high-frequency wall pressure fluctuations is more challenging than for the moderate-frequency range, as highlighted by Zhao [20]. In order to extend the capability of the modal separation algorithm, this study also examines the low-frequency and high-frequency wall pressure fluctuations, as shown in Figure 5.

Figure 5 showcases the modal separation of wall pressure fluctuations at frequencies f = 10 Hz and f = 10,000 Hz. The hydrodynamics mode calculated by the ANN displays good agreement with the theoretical results for both low-frequency and high-frequency wall pressure fluctuations. Moreover, for the high-frequency range at f = 10,000 Hz, the wavelength becomes too short, and the sensor spacing is too wide to display the complete wavelength with a limited sensor density, as demonstrated in Figure 5b. Nevertheless, the calculated results obtained at the selected test points align well with the theoretical results, thereby further confirming the reliability of the ANN developed in this study.

4. Measurement Data Analysis

To further validate the proposed ANN, wall pressure fluctuations were measured using the conventional phase-array approach [18,20] in a wind tunnel with background noise 76.8 dB at the inflow speed of 80 m/s. A total of 25 sensors were installed on a plate in a multi-spiral configuration. The dimension range of the array used in this study was 0.5 m × 0.5 m. To generate fully developed turbulence, a vortex generator in the form of a beam with dimensions 0.5 m × 0.01 m × 0.01 m (length × width × height) was affixed to the plate, as illustrated in Figure 6. For the measurement of wall pressure fluctuations by pressure field microphones (B&K, 4954A), 384,000 samples were collected by a high-speed collector (NI, America) at a frequency of 96 kHz, with flow speeds ranging from 20 m/s to 80 m/s in increments of 20 m/s. The collection of wall pressure fluctuation data was conducted under a fully developed turbulence field. Generally, Wall fluctuation pressure measurements have some uncertainty caused by the background noise of wind tunnels. However, for this measurement, wall pressure fluctuations measured at 80 m/s all exceeded 120 dB, which were 40 dB higher than the background noise. The uncertainty generated by the background noise is too weak to be considered in this work.

4.1. High-Frequency Effect

Typically, the conventional phase-array technique faces challenges in accurately obtaining the spatial correlation of high-frequency wall pressure fluctuations due to the low sensor array density used for measurements [20]. Figure 7 displays the convective speed of high-frequency wall pressure fluctuations calculated by the ANN based on the measured data from the wind tunnel. The dashed-dotted line labeled ‘Upper = 0.7’ and the dotted line labeled ‘Lower = 0.5’ represent the upper and lower limits of the normalized convective speed, respectively [20,28]. The solid line labeled ‘ANN’ indicates the normalized convective speed obtained by the trained ANN using the approximate measured data. The dotted line labeled ‘Conventional’ represents the calculation results obtained using the conventional phase-array technique.

As depicted in Figure 7, the normalized convective speed obtained through the trained ANN falls within the reference speed range defined by the ‘upper limit’ and ‘lower limit’ across a wide frequency range. On the other hand, the normalized convective speed obtained using the conventional phase-array technique only shows agreement with the reference convective speed in the frequency range below 350 Hz. As discussed by Zhao [20], the conventional phase-array technique is inadequate for calculating the convective speed at high frequencies. However, with the ANN developed in this study, a broad frequency range of convective speed can be accurately calculated, with the capability to reach maximum frequencies of up to 10 kHz. This significantly expands the frequency range of the conventional phase-array technique in measuring the spatial correlation of wall pressure fluctuations.

4.2. Low-Frequency Effect

Due to the limited length of the sensor array used to measure wall pressure fluctuations, the accuracy of calculating the low-frequency convective speed is lower compared to the moderate-frequency range [20]. To address this issue, a Gaussian function with different covariance values ($\sigma$, same as ${\sigma }_s$ in Equation (18)) is utilized in the calculation results, as illustrated in Figure 8. The different covariance values are labeled as ‘$\sigma$’, and the normalized convective speed obtained from approximate wall pressure fluctuations in the frequency range of 1 Hz to 130 Hz is depicted. From

Table 1. Comparison of results under different conditions.

No	Conditions	LLPIF
1	$\sigma$ = 1.5	90 Hz
2	$\sigma$ = 1	80 Hz
3	$\sigma$ = 0.5	55 Hz
4	$\sigma$ = 0.05	15 Hz
5	$\sigma$ = 0.005	5 Hz
6	MPA	5 Hz

and Figure 8, it can be observed that the lower limit of the preset identification frequency (LLPIF) represented by the curve labeled ‘1.0’ decreases, and the effective frequency range of the ANN expands as the covariance value $\sigma$ decreases. A smaller covariance value $\sigma$ results in a wider applicable frequency range for the ANN. However, an excessively small covariance value $\sigma$ can lead to overtraining of the ANN, causing significant data oscillation when using the network to calculate the normalized convective speed of the wall pressure fluctuation. Consequently, this can result in calculated convective speeds beyond the reference range, as observed in Figure 8. The frequency expansion and data convergence for convective speed calculation are inherently contradictory and cannot be adequately resolved solely by controlling the covariance value $\sigma$, as demonstrated in Figure 8.

To enhance the performance of the developed ANN, the multi-parameter average (MPA) method is proposed to reduce the LLPIF and mitigate data oscillation caused by overtraining of the ANN. In the MPA method, the output results from the ANN are averaged for different covariance values $\sigma$, specifically at a designated frequency of interest. The procedure for implementing the MPA method can be outlined as follows:

$$r_{OUT}=\left[r_{OUT-1},r_{OUT-2},\cdots ,r_{OUT-K},\right]$$

(22)

where $r_{OUT-i}$ represents the output corresponding to different covariance values $\sigma$, and K represents the total number of covariance values used in the calculation. The mean output can then be expressed as:

$$r_{mean}={\displaystyle \frac{\sum _{i=1}^K{r}_{OUT-i}}{K}},$$

(23)

where $r_{mean}$ represents the calculation result obtained through the MPA method. The MPA method leverages the output results of the ANN for different covariance values $\sigma$. As per Equation (23), the accuracy of the MPA method relies on the number of effective samples obtained at different covariance values $\sigma$. In general, the higher the number of effective samples, the more accurate the calculation results obtained using the MPA method.

As shown in Table 1, the LLPIF achieved by employing the MPA method reaches 5 Hz, which is comparable to the LLPIF attained by using a small covariance value. Additionally, the calculated convective speed obtained through the ANN falls within the reference frequency range across a wide frequency range, as illustrated in Figure 9. Therefore, the MPA method effectively expands the applicable frequency range of the ANN in the low-frequency range and improves the data convergence of the ANN when calculating the spatial correlation of wall pressure fluctuations. At present, the developed ANN demonstrates the capability to accurately predict the spatial correlation of both low-frequency and high-frequency wall pressure fluctuations, as depicted in Figure 9. The calculation accuracy of the normalized convective speed is significantly enhanced compared to that of the conventional phase-array technique.

4.3. Flow Velocity Effect

The flow velocity is an crucial factor that influences the calculation of convective speed based on the measured wall pressure fluctuations. To better understand the impact of flow velocity on spatial correlation calculation, the Artificial Neural Network (ANN) is employed to determine the convective speed across various flow conditions. Figure 10 exhibits the calculated convective speed of wall pressure fluctuations across a wide range of frequencies at different flow velocities.

The LLPIF remains constant at 5 Hz for all the curves, and the normalized convective speed curves display a consistent trend of shifting toward higher values as the flow rate increases, confirming the findings of Zhao [20]. This observation further strengthens the reliability of the ANN in calculating the spatial correlation of wall pressure fluctuations. Moreover, the normalized convective speed under different flow rates exhibits a similar pattern across different frequencies of interest. Initially, the convective speed curves consistently rise with increasing frequency, eventually reaching a plateau. The turning points of the curves occur at 10 Hz, 20 Hz, 30 Hz, and 40 Hz, corresponding to flow rates of 20 m/s, 40 m/s, 60 m/s, and 80 m/s, respectively. To shed light on the mechanism behind the rightward shift in the normalized convective speed curves with increasing flow rate, this study introduces the standard wavelength ${\lambda }_0$, defined as:

$${\lambda }_0={\displaystyle \frac{U_0}{f}},$$

(24)

where $U_0$ represents the freestream velocity and f denotes the frequency of interest.

By substituting the frequency of interest with the standard wavelength, the normalized convective speed depicted in Figure 10 is represented again in Figure 11. The curves of the normalized convective speed for different inflow speeds exhibit a high degree of overlap, particularly at the point where ‘${\lambda }_0=4l_0$’, where $l_0$ corresponds to the length of the sensor array in the direction of flow. This finding suggests that the effective frequency range of the developed ANN is closely linked to the length of the sensor array along the flow direction, an intriguing observation obtained in this study. The ANN can accurately calculate the normalized convective speed when ${\lambda }_0>4l_0$, while significant errors arise when ${\lambda }_0<4l_0$. To elucidate this phenomenon, Figure 12 depicts the wavelengths of the wall pressure fluctuations at various frequencies compared to different standard wavelengths. It can be observed that, when ${\lambda }_0=4l_0$, one quarter of the wall pressure fluctuation wavelength can be captured by the sensor array, and by utilizing wave symmetry, the entire wavelength can be determined. However, as the standard wavelength ${\lambda }_0$ exceeds $4l_0$, the complete wavelength information cannot be obtained, as evidenced in Figure 11, resulting in a larger error in the calculated convective speed of the wall pressure fluctuation. Thus, it can be concluded that the lower frequency limit of the ANN is determined by four times the length of the sensor array along the flow direction.

Furthermore, the impact of high frequencies can also be explained through the aforementioned analysis. In the case of high-frequency wall pressure fluctuations, the sensor spacing becomes too wide, and the sensor density becomes insufficient to capture the complete information of a full-length wave. This limitation sets the upper threshold for the frequency of interest. This phenomenon is clearly demonstrated in Figure 5b, where the sparse distribution of sensors leads to oscillations in the calculation of the normalized convective speed for high-frequency wall pressure fluctuations.

4.4. Sensor Distribution Effect

The accuracy of the spatial correlation calculation is influenced by the distribution pattern of the sensors employed for analysis. To investigate the impact of sensor distribution on the calculation, four distinct sensor distributions are selected from the original sensor array, as depicted in Figure 13. These different sensor distributions consist of 15 sensors each and include linear distribution, circular distribution, cross-distribution, and sparse distribution. In Figure 13, the selected sensors are denoted by square symbols enclosed within dotted lines, while the unselected sensors are represented by circular symbols.

To emphasize the normalized convective speed ($r=U_c∕U_0)$, the identification results are presented as $r-r_{ref}$, where the calculation method is as follows:

$$r-r_{ref}=\left\{\begin{array}{c}r-0.7,\\ 0,\\ r-0.5,\end{array}\begin{array}{c} r>0.7\\ 0.5\le r\le 0.7,\\ r<0.5\end{array}\right.$$

(25)

where $r-r_{ref}$ represents the deviation of the normalized convective speed from the reference range. The normalized convective speeds obtained using different sensor array configurations are illustrated in Figure 14. In Figure 14, the x-axis represents the frequency range, the y-axis represents different array shapes, and the z-axis represents $r-r_{ref}$. The red dotted line on the graph indicates $r-r_{ref}$ = 0, indicating that the normalized convective speed is within the reference range. For comparison purposes, the calculation results based on the original array, as shown in Figure 4, are also included and labeled as “Multispiral” in Figure 14.

Examining Figure 14, it can be observed that the calculation accuracy diminishes when employing different sensor distribution forms compared to the original array. The primary factor contributing to this decline is the reliance of the proposed ANN on the wall pressure fluctuation signals obtained from the sensor arrays. Insufficient sensors compromise the robustness of the ANN method. Regarding the calculation of low-frequency wavenumbers, the results obtained using various sensor distributions exhibit slight deviations from each other, particularly in the case of the circular distribution, as illustrated in Figure 14. The reduced error in the circular array can be attributed to the sufficient array length achieved through sensors distributed along the perimeter, which notably impact the low-frequency calculation [20]. On the other hand, for the medium to high-frequency range, the sparse array demonstrates lower calculation accuracy, as evident in Figure 14. This discrepancy can be attributed to the wide spacing of the sensor array along the flow direction, which is insufficient to capture all the necessary information of a complete wavelength, as discussed in Section 4.3. However, this expanded frequency range can be improved by utilizing a sensor array that incorporates variable spacing and a larger size [18].

4.5. Dead Pixels Effect

Generally, the presence of invalid data measured by malfunctioning sensors is a major cause of large errors in convective speed calculations and should be excluded from the data samples. The measured wall pressure fluctuations in the time domain are depicted in Figure 15. It is evident that two channels with invalid data, exhibiting higher amplitudes compared to others, can be identified in Figure 15a. Following the removal of the invalid data from the samples, the measured wall pressure fluctuations in the time domain are shown in Figure 15b. Utilizing the measured wall pressure fluctuations shown in Figure 15a,b, the calculated convective speed of the wall pressure fluctuation is presented in Figure 16. In Figure 16, the solid line labeled “all the test points included” represents the calculated convective speed.

Measured wall pressure fluctuations, including the two channels with invalid data. The dotted line labeled “with invalid data removed” represents the results obtained using the measurement data with the invalid data removed. The presence of invalid data from malfunctioning test points can introduce errors in certain frequency ranges, and these data points should theoretically be excluded from the effective measurement data. However, in reality, the exclusion of malfunctioning test points does not always result in significant differences in the calculation results. This finding confirms the robustness of the proposed ANN in this study.

5. Conclusions

To address the limitations of conventional phase-array techniques in obtaining spatial correlation of wall pressure fluctuations over a broad frequency range, this study introduces an Artificial Neural Network (ANN) for calculating convective speed representing the spatial correlation of wall pressure fluctuations and expanding the applicable frequency range of the conventional method. The key findings are summarized as follows:

(1)

The developed ANN proves effective in calculating the convective speed at high frequencies. By using the multi-point analysis (MPA) technique on the measurement data before calculating the convective velocity, the frequency applicability and data convergence of the ANN method can be significantly improved.

(2)

The low-frequency limit of the ANN is determined by four times the length of the sensor array in the flow direction, while the calculation error for high frequencies is primarily influenced by the sensor spacing of the sensor array. The higher the accuracy of the measurement data, the better the convergence of the ANN calculation.

(3)

The accuracy of flow velocity calculation slightly decreases when using the sensor distribution form different from the original sensor array. This reduction is generated by the decrease in sensor utilization, leading to a decrease in the robustness of the ANN method.