2. Experimental set-up and flow conditions

2.1. Test facility and flow conditions

The experiments were conducted in the anechoic wind tunnel at Zhejiang University. The exit of the test section is $0.4 \times 0.5$ m $^2$ . The maximum velocity is 60 m s⁻¹, with turbulent intensity below 0.5 %. A NACA0012 aerofoil of 200 mm chord ( $c$ ) and 400 mm span was installed at zero angle of attack along the mid-plane of the test section (figure 1 a). A boundary layer trip of 0.8 mm height was installed at 8 mm downstream of the leading edge on both sides of the aerofoil to obtain a fully developed TBL. The origin of the coordinate system ( $o$ ) is fixed at the centre of the trailing edge. The $y$ and $z$ axes denote the wall-normal and spanwise directions, respectively. The $x$ axis is normal to them following a right-handed system.

Table 1. Summary of the boundary layer properties at $x/c=-0.17,-0.15,-0.13$ .

Figure 1. Model installation and experimental set-up: (a) installation of NACA0012 aerofoil and tomo-PIV set-up; (b) field of view for planar PIV and tomo-PIV; (c) zoomed-in view of the distribution of wall-pressure transducers.

The free-stream velocity was set at $u_\infty =10$ m s⁻¹, resulting in chord-based Reynolds number ( ${\textit{Re}}_c = u_{\infty }c/\nu$ , where $\nu$ is the kinematic viscosity) $1.35 \times 10^5$ . The boundary layer parameters at $x/c=-0.17, -0.15, -0.13$ are estimated, including the boundary layer thickness ( $\delta _{99}$ ), displacement thickness ( $\delta ^*$ ), momentum thickness ( $\theta$ ) and shape factor ( $H = \delta ^*/\theta$ ). The friction velocity ( $u_\tau$ ) is determined by the logarithmic fit of the APG TBL profile following Garcia-Sagrado, Hynes & Hodson (Reference Garcia-Sagrado, Hynes and Hodson2006). The inner viscous scale ( $l=\nu /u_{\tau }$ ) is estimated for scaling. The Reynolds numbers based on friction velocity ( ${\textit{Re}}_\tau = u_{\tau }\delta _{99}/\nu$ ) and momentum thickness ( ${\textit{Re}}_\theta = u_{\infty }\theta /\nu$ ) are estimated. The Clauser pressure-gradient parameter ( $\beta_c=(\theta/\tau_w)\,\text{d}P_\infty/\text{d}x$ , where $\tau _w$ is the wall shear stress, and $P_\infty$ is the pressure at the boundary layer edge) is calculated. The results are summarised in table 1.

2.2. The PIV and wall-pressure fluctuation measurement

For the PIV measurements, the flow was seeded with water–glycol droplets of approximately 1 $\unicode{x03BC}$ m diameter, illuminated by a high-speed laser sheet (Nd:YLF, 527 nm Beamtech Vlite-Hi-527–50K, 50 mJ $@$ 1 kHz). Tomo-PIV was conducted using four CMOS cameras with cross-alignment (Photron FASTCAM, 1024 $\times$ 1024 pixels, 20 $\unicode{x03BC}$ m pixel $^{-1}$ ), as shown in figure 1(a). The camera sensor size was cropped into 512 $\times$ 768 pixels to increase the sampling frequency ( $f_s$ ) to 10 kHz. An ensemble of 20 000 snapshots was captured, corresponding to 1630 boundary layer turnover time. The measurement domain is $20 \times 6 \times 30$ mm $^3$ ( $x \times y \times z$ ), resulting in digital image resolution 23.95 pixels mm $^{-1}$ . The streamwise range of the domain covers $x/c = (-0.19, -0.09)$ (figure 1 b). The wall-normal range extends until the outer layer. DaVis 10 was used to perform volume calibration, volume reconstruction and cross-correlation. The final interrogation volume is $28 \times 28 \times 28$ voxels with 75 % overlap, resulting in vector pitch 0.29 mm (corresponding to $9l$ ). Some 25 vectors are obtained for resolving the boundary layer thickness 7.11 mm at $x/c = -0.15$ , which is comparable to the resolution typically employed in boundary layer measurements (Ghaemi & Scarano Reference Ghaemi and Scarano2011; Pröbsting et al. Reference Pröbsting, Scarano, Bernardini and Pirozzoli2013; Groot et al. Reference Groot, Serpieri, Pinna and Kotsonis2018). Eight vectors are obtained for the hairpin head, which is higher than the typical PIV measurements requiring 5–6 vectors (Ye, Schrijer & Scarano Reference Ye, Schrijer and Scarano2016; Raffel et al. Reference Raffel, Willert, Scarano, Kähler, Wereley and Kompenhans2018).

To enhance the spatial resolution, the planar PIV was performed using a CMOS camera (Phantom VEO E-340L, 2560 $\times$ 1600 pixels, 10 $\unicode{x03BC}$ m pixel $^{-1}$ ). The sensor is cropped into 512 $\times$ 480 pixels. The field of view is 21.7 $\times$ 19.5 mm $^2$ ( $x\times y$ , figure 1 b), starting from $x/c=-0.20$ . The acquisition frequency and digital image resolution remain the same as in tomo-PIV measurements. An ensemble of 50 000 snapshots was captured, corresponding to 4080 boundary layer turnover time. The premultiplied spectra of streamwise velocity fluctuations using different ensemble sizes are assessed in Appendix A, confirming good statistical convergence. The velocity fields are calculated with sliding-averaged correlation with three sliding pairs. The final window size is $16 \times 16$ pixels with 75 % overlap, resulting in a smaller vector pitch 0.17 mm (corresponding to $5l$ ). Approximately 43 vectors are obtained to resolve the boundary layer, and 15 vectors are obtained to capture the hairpin head.

The uncertainty analysis of the mean streamwise velocity ( $\varepsilon$ ) and the standard deviation of streamwise velocity ( $\varepsilon _\sigma$ ) at $y^+ = 50$ using maximum ensemble size is performed following Ye et al. (Reference Ye, Schrijer and Scarano2016) and Sciacchitano & Wieneke (Reference Sciacchitano and Wieneke2016). The effective number of snapshots for a time-resolved dataset is calculated, which is 3537 and 1535 for planar PIV and tomo-PIV, respectively, based on the temporal autocorrelation coefficient of the measured signal (Sciacchitano & Wieneke Reference Sciacchitano and Wieneke2016; Manovski et al. Reference Manovski, Abu Rowin, Ng, Gulotta, Giacobello, De Silva, Hutchins and Marusic2025). The error related to reconstruction and cross-correlation algorithm ( $\varepsilon _{cc}$ ) is 0.3 voxels for tomo-PIV (Lynch & Scarano Reference Lynch and Scarano2015), and 0.2 pixels for planar PIV (Ghaemi, Ragni & Scarano Reference Ghaemi, Ragni and Scarano2012). The non-dimensional results are summarised in table 2.

Table 2. Uncertainty parameters of 20 000 and 50 000 snapshots for tomo-PIV and planar PIV, and 100 000 samples for the wall-pressure transducer. Here, $\varepsilon _{cc}$ is the uncertainty related to reconstruction and cross-correlation, $\varepsilon$ represents the uncertainty of the mean value, and $\varepsilon _{\sigma }$ is the uncertainty of the standard deviation.

Wall-pressure fluctuations were measured simultaneously with both tomo-PIV and planar PIV using three miniature Knowles FG-23329-P07 transducers (frequency response $\pm3$ dB, frequency range 100 Hz to 10 kHz). In-house calibration was performed for these transducers using a GRAS 40PH-10 CCP far-field microphone following Ali (Reference Ali2018) and Pan & Ye (Reference Pan and Ye2025). The resultant frequency response exhibits a smooth sensitivity with deviation less than 2.5 dB, and a slight phase shift below 0.2 rad across a broadband frequency range from 50 Hz to 3 kHz. The transducers were mounted at $x/c =-0.17, -0.15, -0.13$ along $z = 0$ , as shown in figure 1(c). They are mounted with a pinhole configuration of 0.4 mm diameter to avoid high-frequency attenuation. The data were acquired at $f_s = 40$ kHz for 10 s, corresponding to 8160 boundary layer turnover time. The wall-pressure fluctuations are then down-sampled to 10 kHz to perform the $p$ – $u$ correlation analysis. The uncertainties based on the ensemble size of 100 000 samples (with effective number 23 510) are estimated and summarised in table 2.

2.3. The EMD

As an a posteriori adaptive method for scale decomposition, EMD extracts the IMFs of different scales that are embedded within the original signal $f(t)$ (Huang et al. Reference Huang, Shen, Long, Wu, Shih, Zheng, Yen, Tung and Liu1998). The decomposition is driven by iteratively subtracting the average of the upper and lower envelopes of $f(t)$ until only a residual $r(t)$ remains. The detailed steps for the iterative progress are as follows.

1. (i) The residual $r(t)$ is defined and initialised by $f(t)$ in the beginning.

2. (ii) The local maxima and minima peaks of $r(t)$ are detected. If no maxima or minima peaks are found, i.e. $r(t)$ is monotonic, then the iterative decomposition progress is terminated.

3. (iii) The maxima and minima peaks are connected and fitted with a cubic spline function to generate the upper and lower envelopes ( $E_{\textit{up}}(t)$ and $E_{\textit{low}}(t)$ , respectively) (Cheng et al. Reference Cheng, Li, Lozano-Durán and Liu2019; Mäteling et al. Reference Mäteling, Klaas and Schröder2020).

4. (iv) The residual $r(t)$ is updated by subtracting the average of the upper and lower envelopes from the residual of the previous iteration as

   (2.1) \begin{equation} r_k(t) = r_{k-1}(t)-\bigl(E_{\textit{up}}(t)+E_{\textit{low}}(t)\bigr)/2. \end{equation}

5. (v) The convergence criterion is evaluated based on the mean square difference

   (2.2) \begin{equation} \textit{SD} = \frac {\sum |r_k(t)-r_{k-1}(t)|^2}{\sum r^2_{k-1}(t)}. \end{equation}
   Following Huang et al. (Reference Huang, Shen, Long, Wu, Shih, Zheng, Yen, Tung and Liu1998), the iteration (steps (ii)–(v)) is stopped if $SD\leqslant 0.1$ . The first IMF is defined as $\text{IMF}^f_1(t)=r_k(t)$ when reaching the convergence criterion.

6. (vi) The input signal $f(t)$ is updated by subtracting the obtained $\text{IMF}(t)$ . Steps (i)–(v) are repeated to extract the higher-order IMFs.

The original signal is eventually decomposed to a series of IMFs and a final residual as

(2.3) \begin{equation} f(t)=\sum _{i=1}^{N}\text{IMF}^f_{i}(t)+r(t), \end{equation}

where $i$ is the order of the $\text{IMF}$ s. The length scale of each $\text{IMF}^f_i$ increases with its order. The superscript $f$ denotes the original signal being decomposed, and $N$ is the total number of decomposed IMFs.

3. Global flow characteristics

The time-averaged velocity profiles at $x/c=-0.15$ are plotted in semi-log scale in figure 2(a), where $u^+ = \overline {u}/u_\tau$ , $\overline {u}$ is the time-averaged streamwise velocity, and $y^+ = yu_\tau /\nu$ . The profile of ZPG TBL at ${\textit{Re}}_\tau =256$ is included for comparison. This TBL was obtained by an additional measurement on a flat plate, whose detailed parameters can be found in the previous research of Feng & Ye (Reference Feng and Ye2023). For the APG condition, profiles obtained from planar PIV and tomo-PIV collapse in the range $y^+=(25, 200)$ . The planar PIV result covers a broader wall-normal range, from the buffer layer to the free-stream. Due to the non-equilibrium effect in the APG TBL, the log layer exhibits a steeper slope ( $\kappa = 0.3$ , $B = 1.4$ ) than the ZPG condition ( $\kappa = 0.41$ , $B = 5.0$ ) (Ghaemi & Scarano Reference Ghaemi and Scarano2011).

Figure 2(b) shows the Reynolds stress ( $\overline {u'u'}$ , $\overline {v'v'}$ and $\overline {u'v'}$ , where $u'$ and $v'$ are the streamwise and wall-normal velocity fluctuations, respectively). All components show good agreement for planar and tomographic PIV measurement results. Compared with the ZPG condition, $\overline {u'u'}$ exhibits a higher value in the log region and lower wake region ( $y^+ = [40, 100]$ ), confirming the energisation of turbulent structures under APG (Tanarro et al. Reference Tanarro, Vinuesa and Schlatter2020).

Figure 2. Semi-log profile of (a) the time-averaged streamwise velocity and (b) the Reynolds stress at $x/c = -0.15$ . Reference results of the ZPG TBL are included for comparison, satisfying the logarithmic fit of a fully developed ZPG TBL (White Reference White2006).

The Clauser pressure-gradient parameter ( $\beta _c$ ) along the entire streamwise range is evaluated following the definition of Clauser (Reference Clauser1954), with results presented in figure 3. The $\beta _c$ value increases moderately from 0.63 to 1.48 over $x_l/c=0.81-0.91$ (where $x_l$ is the chordwise distance from the leading edge). The mild $\beta _c$ within the TBL near the NACA0012 trailing edge agrees well with that reported by Tanarro et al. (Reference Tanarro, Vinuesa and Schlatter2020), which is due to its modest camber.

Figure 3. The Clauser pressure-gradient parameter along the entire streamwise range ( $\beta _c \sim x_l/c$ ) of the measurement domain.

A TBL exhibits sufficient scale separation when it is well behaved (Dróżdż et al. Reference Dróżdż, Elsner and Drobniak2015; Vinuesa et al. Reference Vinuesa, Örlü, Sanmiguel Vila, Ianiro, Discetti and Schlatter2017b ). The state of the present APG TBL is characterised using the diagnostic-plot scaling following Vinuesa et al. (Reference Vinuesa, Negi, Atzori, Hanifi, Henningson and Schlatter2018). The scaled root mean square profiles of streamwise velocity fluctuations ( $u'_{rms}/(\overline {u}/\sqrt {H})$ ) are represented as functions of mean streamwise velocity ( $\overline {u}/u_e$ , where $u_e$ is the velocity at the boundary layer edge), as shown in figure 4. The diagnostic plots exhibit linear behaviour encompassing the range $\overline {u}/u_e=0.8{-}0.9$ , indicating that the APG TBL has reached a well-behaved state. Consequently, sufficient scale separation is achieved for scale-based decomposition.

Figure 4. Diagnostic plot scaling modified with the shape factor $H$ of the trailing-edge TBL.

The power spectrum densities (PSD $\varPhi _{pp}(\omega )$ , $\omega =2\pi f$ ) of wall-pressure fluctuations in the APG TBL close to the trailing edge ( $x/c=-0.15$ , ${\textit{Re}}_{\tau }=216$ ) and ZPG TBL over the flat plate ( ${\textit{Re}}_{\tau }=256$ ) are estimated. The PSD are scaled using $U_e$ (velocity at boundary layer edge), $\tau _w$ and $\delta ^*$ following Goody (Reference Goody2004) and Lee (Reference Lee2018), as shown in figure 5. The wall-pressure spectrum in the APG TBL exhibits broadband energy elevation compared with the ZPG condition (Rozenberg et al. Reference Rozenberg, Robert and Moreau2012). The PSD of ZPG condition exhibit decay rates $\omega ^{-0.78}$ and $\omega ^{-5}$ in the mid- and high-frequency bands, respectively. Faster decay rates $\omega ^{-1.8}$ and $\omega ^{-7.9}$ are observed for the APG conditions in the former frequency bands, respectively. The decay rates agree well with the empirical wall-pressure fluctuations (Goody Reference Goody2004; Lee Reference Lee2018), validating the current wall-pressure measurements.

Figure 5. The scaled PSD and empirical models of the wall-pressure fluctuations of trailing edge APG and flat plate ZPG TBLs.

4. Instantaneous flow organisation

The three-dimensional instantaneous flow organisation and dominant coherent structures are shown in figure 6 (see also supplementary movie 1). The velocity streaks are visualised by the iso-surface of $|u'|/u_\infty = 0.1$ . The vortical structures are detected by the $Q$ -criterion with iso-value $0.2 \times 10^6$ s $^{-2}$ . The low- and high-speed streaks (blue and red iso-surfaces) appear alternately in the spanwise direction, with lengths up to $600l$ (figure 6 $a$ ). Large-scale vortices are populated over low-speed streaks (figure 6 $b$ ) (Adrian Reference Adrian2007), featuring a full hairpin shape and its derivatives of quasi-streamwise tube and cane shapes. Zoomed-in views of the former vortical structures are shown in figures 6( $b_1$ – $b_3$ ), superposed with the cross-plane ( $x{-}y$ ) contour of velocity fluctuations and vector fields. Their lengths fall in the range of $100l$ to $200l$ . The sweep and ejection motions induced by the spanwise and streamwise components of hairpin-like structures (Adrian Reference Adrian2007) are clearly observed in the corresponding cross-plane vector fields (figures 6 $b$ 1– $b$ 3). Due to the accumulation and auto-generation of hairpin-like structures, hairpin packets with three large-scale vortices also form, as shown in figure 6( $b$ 4). The vector field indicates that a shear layer is formed due to the sequential sweep and ejection motions induced by adjacent hairpin-like vortices. The overall scenario agrees with the previous measurement of the trailing edge TBL of an NACA0012 aerofoil reported by Ghaemi & Scarano (Reference Ghaemi and Scarano2011).

Figure 6. Instantaneous flow organisation. The low- and high-speed streaks are visualised by the blue and red iso-surfaces of $|u'|/u_\infty =0.1$ , respectively. The vortical structures (yellow iso-surfaces) are identified by $Q=0.2\times 10^{6}\ \text{s}^{-2}$ : (a) low- and high-speed streaks; (b) vortical structures and low-speed streaks. Zoomed-in views of (b1) hairpin, (b2) cane, (b3) quasi-streamwise vortex, and (b4) hairpin packet. The $x{-}y$ cross-plane contour with vectors of velocity fluctuations is superposed in each zoomed-in view.

5. Scale identification of turbulent structures and wall pressure

To isolate the turbulent structures and wall-pressure fluctuations of different scales embedded in the original data, 1-D EMD is applied to the velocity and wall-pressure fluctuations ( $u'(t)$ , $v'(t)$ , $w'(t)$ and $p'(t)$ ) at $x/c=-0.15$ . The IMFs at different scales (IMF $^u$ , IMF $^v$ , IMF $^w$ and IMF $^p$ , respectively) are extracted. The premultiplied spectra of both original and IMFs of streamwise velocity fluctuations ( $\varPhi ^+_{uu}/\lambda ^+_x$ , where $\lambda ^+_x$ is the inner-scaled streamwise wavelength) and wall-pressure fluctuations ( $\varPhi ^+_{pp}/\lambda ^+_x$ ) are estimated at $x/c = -0.15$ . The streamwise wavelength is converted from the frequency by considering the Taylor hypothesis (Gibeau & Ghaemi Reference Gibeau and Ghaemi2021; Baars, Dacome & Lee Reference Baars, Dacome and Lee2024), and scaled with inner viscous scales. The $\varPhi ^+_{uu}$ and $\varPhi ^+_{pp}$ values are scaled using $u_\tau ^2$ and $\rho ^2u_\tau ^4$ (where $\rho$ is the fluid density), following Ghaemi & Scarano (Reference Ghaemi and Scarano2013). The results are shown in figures 7, 8 and 9. The EMD-based scale decomposition was also performed at two higher Reynolds number conditions ( ${\textit{Re}}_c=2.03\times 10^5$ and $2.70\times 10^5$ ) for comparison. The detailed results are provided in Appendix B.

Figure 7. Premultiplied spectra of streamwise velocity fluctuations ( $\varPhi ^+_{uu}/\lambda ^+_x$ ) at $x/c=-0.15$ estimated using data measured by (a) planar PIV and (b) tomo-PIV, respectively. The spectral peaks are denoted by magenta circles.

Figure 8. The premultiplied spectra ( $\varPhi ^+_{uu}/\lambda ^+_x$ ) of decomposed ( $a$ , $c$ , $e$ , $g$ , $i$ ) $\text{IMF}^u_1$ – $\text{IMF}^u_5$ from planar PIV and ( $b$ , $d$ , $f$ , $h$ , $j$ ) $\text{IMF}^u_2$ – $\text{IMF}^u_6$ from tomo-PIV at $x/c=-0.15$ . The spectral peaks are denoted by magenta circles.

Figure 9. The premultiplied spectrum of original $p'$ and decomposed $\text{IMF}^p$ s obtained from the transducer at $x/c=-0.15$ , with the maxima denoted by squares.

The inner-scaled premultiplied spectra of original streamwise velocity fluctuations from planar PIV and tomo-PIV (figure 7) exhibit similar patterns, with high-energy inner peaks $\lambda _x^+ = 740$ and 720, respectively, at $y^+ \lt 20$ , which are attributed to the quasi-streamwise vortices and velocity streaks (Wang, Hu & Zheng Reference Wang, Hu and Zheng2021). The spectrum from tomo-PIV (figure 7 $b$ ) exhibits high-frequency noise at $\lambda _x^+ \lt 100$ ( $f \gt 1900$ Hz) (Wernet Reference Wernet2017). The former frequency range is considered irrelevant for the present analysis.

After applying EMD, the IMF $^u$ with increasing scales are obtained. For the IMF $^u$ from planar PIV, the inner-scaled premultiplied spectra peak at $\lambda ^+_{peak}=310$ , 630, 1170, 2350 and 5470, as shown in figures 8( $a$ ), 8( $c$ ), 8( $e$ ), 8( $g$ ), and 8( $i$ ). The high-energy range of the premultiplied spectrum moves closer to the wall as the order of IMF $^u$ increases. Due to the alternating distribution of positive and negative structures, the resultant length of an individual structure is half of $\lambda ^+_{peak}$ (Gibeau & Ghaemi Reference Gibeau and Ghaemi2021). The half-lengths of the first three IMF $^u$ are consistent with the sizes of hairpins, hairpin packets and velocity streaks observed in figure 6, respectively. For IMF $^u_4$ and IMF $^u_5$ , the half of $\lambda ^+_{peak}$ exceeds $5.4\delta$ , corresponding to the elongated near-wall streaks (Jiménez et al. Reference Jiménez, Del Álamo and Flores2004). The former structures are not dominant at the present low Reynolds number condition. For IMF $^u_1$ from tomo-PIV, the smallest wavelength $40l$ corresponds to the measurement noise of tomo-PIV, which comes from the volume reconstruction from projected particle images obtained by a finite number of cameras (Lynch & Scarano Reference Lynch and Scarano2014). Consequently, its premultiplied spectrum is not presented. For higher-order modes, the spectra of IMF $^u_2$ to IMF $^u_5$ (figures 8 b,d, f,h) and the peak wavelength from tomo-PIV exhibit good agreement with those of IMF $^u_1$ to IMF $^u_4$ from the planar PIV result. The IMFs of wall-normal velocity fluctuations exhibit characteristic wavelengths similar to those of the streamwise component (see table 3) and are not elaborated here.

Table 3. Characteristic wavelengths of the original signal and IMFs for $u'$ , $v'$ and $p'$ determined at $x/c=-0.15$ .

For the wall-pressure fluctuations, the spectrum of the original signal peaks at $\lambda _{peak}^+= 280$ (figure 9, black line), which is smaller than the value for the streamwise velocity fluctuations. This observation aligns with findings reported by Baars et al. (Reference Baars, Dacome and Lee2024) and Hassanein et al. (Reference Hassanein, Modesti, Scarano and Baars2024) at ${\textit{Re}}_{\tau }= 550{-}5200$ . The same trend is also evident for the $\text{IMF}^p$ and $\text{IMF}^u$ with the same orders, as shown in table 3.

The energy content ( $k$ ) of each IMF $^u$ or IMF $^p$ is defined following Sun & Yan (Reference Sun and Yan2017) as

(5.1) \begin{equation} k=\frac {\int {(\text{IMF}_i})^2\,{\rm d}t}{\sum _{i=1}^N\int {(\text{IMF}_i})^2\,{\rm d}t+\int {r^2\,{\rm d}t}}. \end{equation}

The energy contents of the IMF $^p$ are estimated at $x/c=-0.15$ . Those of the IMF $^u$ are estimated at selected wall-normal locations $y^+=25$ , 41, 72 and 134 above the former transducer measured by planar PIV. The results are shown in figure 10. For the IMF $^u$ (figure 10 $a$ ), IMF $^u_2$ exhibits the highest energy level of over 30 %. Considering the wavelength, this indicates the energy dominance of hairpin packets. The first three IMF $^u$ occupy energy more than 78.4 % in total. The high-order IMF $^u$ are considered insignificant in the present analysis. For the IMF $^p$ (figure 10 $b$ ), the IMF $^p_1$ and IMF $^p_2$ with smaller wavelengths show substantial energy contributions, accounting for 41.0 % and 34.3 % in the total. This is attributed to the intense burst–sweep events in the high-frequency component of wall-pressure fluctuations (Thomas & Bull Reference Thomas and Bull1983). The cumulative energy of the first four $\text{IMF}^p$ occupies 94.5 % in total, indicating their dominance.

Figure 10. The energy content of each $\text{IMF}$ at $x/c = -0.15$ : ( $a$ ) streamwise velocity fluctuations from planar PIV at $y^+=25$ , 41, 72 and 134; ( $b$ ) wall-pressure fluctuations.

6. Scale-dependent pressure–velocity correlation

The correlation ( $R_{p_mu_n}$ ) between the streamwise velocity and wall-pressure fluctuations at different scales is estimated using $\text{IMF}^p_m$ and $\text{IMF}^u_n$ at the streamwise location of the reference pressure transducer ( $x/c=-0.15$ ):

(6.1) \begin{equation} R_{p_mu_n} = \frac {1}{\rho u_\infty ^3} \left \langle \text{IMF}^p_m(t) \times\text{IMF}^u_n(y^+, t) \right \rangle. \end{equation}

Accounting for the wavelengths of coherent structures and the energy contributions of corresponding IMFs (see figure 10), the first five IMF $^p$ and IMF $^u$ ( $1\leqslant m, n \leqslant 5$ ) are selected for estimating $R_{p_mu_n}$ . The $R_{p_mu_n}$ values at $y^+=25$ , $41$ , $72$ and $134$ estimated using planar PIV data are shown in figure 11. (Results for higher ${\textit{Re}}_c$ are available in Appendix C.) Significant correlations arise in the buffer layer ( $y^+= 25$ ) and log layer ( $y^+=41$ and $72$ ). This is consistent with the dominant wall-pressure source identified by Garcia-Sagrado & Hynes (Reference Garcia-Sagrado and Hynes2012) for the same aerofoil model. As shown in figures 11(a–c), significant correlations appear at both the diagonal and the off-diagonal condition when $m = n+1$ . The latter exhibits higher $R_{p_mu_n}$ . Since characteristic wavelengths of $\text{IMF}^p_{n+1}$ and $\text{IMF}^u_n$ share the closest similarity (see table 3), it suggests that wall-pressure fluctuations originate from the turbulent structure with a similar scale (Gibeau & Ghaemi Reference Gibeau and Ghaemi2021). This suggests that wall-pressure fluctuations exhibit a linear superposition behaviour with the field coherent structures, analogous to the superposition effect between the inner and outer turbulent motions (Marusic et al. Reference Marusic, Mathis and Hutchins2010a ; Mathis, Hutchins & Marusic Reference Mathis, Hutchins and Marusic2011; Baars, Hutchins & Marusic Reference Baars, Hutchins and Marusic2016). Recent studies of Baars et al. (Reference Baars, Dacome and Lee2024) observed the significant linear frequency coherence between wall-pressure and streamwise velocity fluctuations in the frequency domain for TBL, ${\textit{Re}}_{\tau } = 550{-}5200$ , which also indicates such a superposition effect. Moving away from the wall, $R_{p_mu_n}$ becomes moderate in the wake region ( $y^+ = 134$ , figure 11 $d$ ), indicating weak correlation. The $R_{p_mu_n}$ vakue based on tomo-PIV measurement also exhibits significant correlation when IMF $^u_n$ and IMF $^p_m$ share similar scales ( $m=n$ due to the existence of noise-related IMF $^u_1$ as shown in table 3), which is not elaborated here for brevity.

Figure 11. Correlations between IMFs of streamwise velocity from planar PIV and wall-pressure fluctuations ( $R_{p_mu_n}$ ): (a–d) $y^+=25$ , 41, 72 and 134, respectively.

To figure out the spatial distribution of the pressure sources at each strongly correlated wavelength ( $m=n+1$ and $m = n$ for two- and three-dimensional velocity fields, respectively), the spatial–temporal correlations ( $p$ – $u$ , $p$ – $v$ and $p$ – $w$ ) are estimated using the extracted IMFs of the corresponding orders following

(6.2) \begin{equation} \left . \begin{array}{ll} \displaystyle R^{2\text{-}D}_{p_{n+1}U_n}(x,y) = \frac {1}{\rho u_\infty ^3} \left \langle \text{IMF}^p_{n+1}(t) \times\text{IMF}^{U}_n(t,x,y) \right \rangle ,\\[10pt] \displaystyle R^{3\text{-}D}_{p_{n}U_n}(x,y,z) = \frac {1}{\rho u_\infty ^3} \left \langle \text{IMF}^p_{n}(t) \times\text{IMF}^{U}_n(t,x,y,z) \right \rangle , \end{array}\right \} \end{equation}

where $U$ denotes $u'$ , $v'$ and $w'$ . The superscripts $2\text{-}D$ and $3\text{-}D$ mean that the IMF $^U$ come from planar PIV and tomo-PIV measurement, respectively. Positive and negative $\text{IMF}^p$ are considered separately in (6.2) to isolate the related turbulent structures. The resultant scale-dependent spatial–temporal $p$ – $u$ correlations are denoted as $R^{2-D+}_{p_{n+1}U_n}$ , $R^{2-D-}_{p_{n+1}U_n}$ , $R^{3-D+}_{p_nU_n}$ and $R^{3-D-}_{p_nU_n}$ , where superscript $+$ and $-$ indicate the sign of IMF $^p$ .

Figure 12. Spatial distribution of $R^{2D}_{p_{n+1}u_n}$ : ( $a, d$ ) $n = 1$ , ( $b, e$ ) $n = 2$ , ( $c, f$ ) $n = 3$ , for (a–c) $R^{2D+}_{p_{n+1}u_n}$ , (d–f) $-R^{2D-}_{p_{n+1}u_n}$ . The vector fields of the $p$ – $u$ and $p$ – $v$ correlations are superimposed. The magenta dashed line denotes the location of the reference transducer.

The contours of the $p$ – $u$ correlation estimated from planar PIV data ( $R^{2D}_{p_{n+1}U_n}$ ) are shown in figure 12. The vector field associate with $R^{2D}_{p_{n+1}U_n}$ is superimposed. The correlations of negative $\text{IMF}^p_{n+1}$ are multiplied by $-1$ to represent the exact flow direction (Gibeau & Ghaemi Reference Gibeau and Ghaemi2021). Since the first four $\text{IMF}^p$ occupy the dominant energy (figure 10), only $R^{2D}_{p_{2}u_{1}}$ , $R^{2D}_{p_{3}u_{2}}$ and $R^{2D}_{p_{4}u_{3}}$ are discussed. The correlation pattern for $R^{2D}_{p_{1}u_{1}}$ is similar to that for $R^{2D}_{p_{2}u_{1}}$ with a smaller length scale. The former is not discussed here for conciseness.

For $R^{2D}_{p_{2}u_{1}}$ (figures 12 a,d), alternating positive and negative correlation regions with lengths approximately $200 l$ and $100 l$ , respectively, arise around the reference transducer. The vector field indicates that the positive wall-pressure fluctuations ( $R^{2D+}_{p_{2}u_{1}}$ , figure 12 $a$ ) occur when the upstream sweep and downstream ejection motions impinge, forming a stagnated shear layer, which is related to the turbulent motions induced between two adjacent hairpin-like vortices as observed in the three-dimensional instantaneous fields (figure 6 $b$ 4). The negative wall-pressure fluctuations ( $-R^{2D-}_{p_{2}u_{1}}$ , figure 12 $d$ ) correspond to the fluid splitting due to upstream ejection and downstream sweep motions, which occur around the spanwise component of hairpins (figure 6 $b$ 1) and canes (figure 6 $b$ 2). This is consistent with the findings reported by Gibeau & Ghaemi (Reference Gibeau and Ghaemi2021), who demonstrated that the sweep and ejection motions induced by hairpins contribute to the mid-frequency wall-pressure fluctuations. The intermittent large-scale sweep and ejection motions also induce positive and negative curvatures of the velocity profiles (Ghaemi & Scarano Reference Ghaemi and Scarano2013), which lead to the increase of the mean shear and drive the rapid pressure term (Kraichnan Reference Kraichnan1956). For $R^{2D}_{p_{3}u_{2}}$ (figures 12 b,e), the alternating correlation regions extend in the streamwise direction, yielding a larger length, approximately $300l$ . The organisations of correlation patterns exhibit similarity with those of $R^{2D}_{p_{2}u_{1}}$ , indicating the self-similarity between these two structures (Marusic & Monty Reference Marusic and Monty2019; Baars & Marusic Reference Baars and Marusic2020; Gibeau & Ghaemi Reference Gibeau and Ghaemi2021). The pressure source corresponds to the uniform momentum zones (UMZs) enclosing multiple hairpin packets (Adrian, Meinhart & Tomkins Reference Adrian, Meinhart and Tomkins2000; Adrian Reference Adrian2007). For $R^{2D}_{p_{4}u_{3}}$ , a single positively correlated region with length over $500l$ is observed upstream of the reference transducer (figures 12 c,f). The corresponding height is reduced by 50 % compared with heights of $R^{2D}_{p_{2}u_{1}}$ and $R^{2D}_{p_{3}u_{2}}$ , with the wall-normal component weak. As a result, the positive and negative IMF $^p_4$ are related to the high- and low-speed velocity streaks (figure 6 $a$ ), being accompanied by weak sweep and ejection motions, respectively. The contribution of elongated velocity streaks to the low-frequency wall-pressure source is also reported by Pan & Ye (Reference Pan and Ye2025) close to the trailing edge of an NACA0018 aerofoil.

Figure 13. Spatial distribution of $R^{3D}_{p_{n}u_n}$ : ( $a$ , $d$ , $g$ , $j$ ) $n =2$ , ( $b$ , $e$ , $h$ , $k$ ) $n =3$ , ( $c$ , $f$ , $i$ , $l$ ) $n =4$ . The three-dimensional distributions of ( $a$ – $c$ ) $R^{3D+}_{p_{n}u_n}$ and ( $g$ – $i$ ) $-R^{3D-}_{p_{n}u_n}$ are shown by red (positive value) and blue (negative value) iso-surface, respectively. Iso-values are set to $|R^{3D}_{p_{n}u_n}|=3.5\times 10^{-6}$ , $2.5\times 10^{-6}$ and $1.5\times 10^{-6}$ for $n=2,3,4$ , respectively. The cross-plane ( $x{-}z$ ) contours of (d–f) $R^{3D+}_{p_{n}u_n}$ and (j–l) $-R^{3D-}_{p_{n}u_n}$ at $y^+=50$ are also shown, superimposed with vector fields of ( $p$ – $u$ , $p$ – $w$ ) correlation. The reference transducer is denoted by the black circles.

Similarly, the three-dimensional spatial distributions of $p$ – $u$ correlations ( $R^{3D}_{p_{n}u_n}$ ) are estimated based on the velocity fields obtained from tomo-PIV measurement, as shown in figure 13. The strongly correlated modes of $m = n$ ( $n=2{-}4$ ) are considered. The three-dimensional correlation of $R^{3D}_{p_2u_2}$ (figures 13 a,g) and $R^{3D}_{p_3u_3}$ (figures 13 b,h) exhibit alternating positive and negative correlation regions arise around the transducer with self-similarity. In particular, correlation vectors in the $x{-}z$ cross-plane at $y^+=50$ show that low-speed motions are flanked by a pair of counter-rotating wall normal vortices (magenta arrows) for $R^{3D}_{p_2u_2}$ (figures 13 d,j). This corresponds to the legs of hairpins (Adrian Reference Adrian2007; Gibeau & Ghaemi Reference Gibeau and Ghaemi2021), which further proves the contribution of hairpin-related vortices to IMF $^p_2$ . Although the streamwise scale of $R^{3D}_{p_3u_3}$ is larger than that of $R^{3D}_{p_2u_2}$ , their spanwise scales are similar. For $R^{3D}_{p_4u_4}$ (figures 13 c,f,i,l), the elongated correlation regions are observed to appear side by side in the spanwise direction from the three-dimensional perspective. This is consistent with the spanwise distribution of the velocity streaks (Jiménez et al. Reference Jiménez, Del Álamo and Flores2004).

7. Scale-dependent conditional average

To further investigate the spatial–temporal evolution of turbulent structures modulating wall-pressure fluctuations of certain scales, the conditional average of original $u'$ and $v'$ is performed based on the occurrence of zero-crossing (ZC) events (Baars et al. Reference Baars, Talluru, Hutchins and Marusic2015) of $\text{IMF}^p_2$ to $\text{IMF}^p_4$ . Considering the larger sampling ensemble size and higher signal-to-noise ratio, the scale-dependent conditional average is performed based on the velocity fields measured by planar PIV. The ZC events represent the change between positive and negative wall-pressure fluctuations. The criterion to detect positive (ZC $^+$ ) and negative (ZC $^-$ ) zero-crossing events are given by

(7.1) \begin{equation} \left . \begin{array}{ll} \displaystyle \text{ZC}^+ \quad \text{if } p' = 0\quad \&\quad {\rm d}p'/{\rm d}t \gt 0,\\ \displaystyle \text{ZC}^- \quad \text{if } p' = 0\quad \&\quad {\rm d}p'/{\rm d}t \lt 0. \end{array}\right \} \end{equation}

An example of the detected events is shown in figure 14(a) for the time series of $\text{IMF}^p_2$ . The events ZC $^+$ and ZC $^-$ are marked with red upward and blue downward triangles, respectively. The adjacent positive and negative pressure peaks (PP $^+$ and PP $^-$ , marked with red circles and blue squares) are also selected as criteria to represent local pressure extrema. The time series resembles a sinusoid signal. The PP $^+$ , ZC $^-$ , PP $^-$ and ZC $^+$ events correspond to phases $\pi /2$ , $\pi$ , $3\pi /2$ and $2\pi$ , respectively. The contours of conditional averaged streamwise velocity fluctuations ( $\overline {u'}/u_{\infty }$ ) based on aforementioned events of $\text{IMF}^p_2$ to $\text{IMF}^p_4$ are shown in figures 14(b–d). The corresponding vector fields are superimposed to visualise the turbulent motions.

Figure 14. (a) Example of the time series of $\text{IMF}^p_2$ (black line) and detected events of PP $^+$ ( $\circ$ ), ZC $^-$ ( $\triangledown$ ), PP $^-$ ( $\square$ ) and ZC $^+$ ( $\triangle$ ). (b–d) Contours of conditional averaged streamwise velocity fluctuations ( $\overline {u'}/u_{\infty }$ ) based on the pressure events of $\text{IMF}^p_2$ , $\text{IMF}^p_3$ and $\text{IMF}^p_4$ . The vector fields are superimposed. Images 1–4 are based on the PP $^+$ , ZC $^-$ , PP $^-$ , and ZC $^+$ events, respectively. The magenta dashed line denotes the location of the reference transducer.

For all the IMF $^p$ , the convection of the dominant high- and low-speed structures is observed, following the phase evolutions of wall-pressure fluctuations. For IMF $^p_2$ (figure 14 b), a high-speed structure of length $200l$ initially appears upstream of the reference transducer, being followed by a weaker low-speed structure for the PP $^+$ event (phase $\pi /2$ , figure 14 b1). The former structures are associated with upstream sweep and downstream ejection motions, forming an impinging shear layer that induces a positive pressure peak. As the phase increases, the high-speed structure evolves downstream, locating directly above the transducer for phase $\pi$ (ZC $^-$ , figure 14 b2). The associated sweep motion becomes dominant, causing a positive-to-negative shift in wall-pressure fluctuations. At phase $3\pi/2$ (figure 14 b3), a weak low-speed motion forms upstream of the high-speed one. The fluid splitting due to upstream ejection and downstream sweep is produced above the transducer, triggering a negative pressure peak (PP $^-$ ). For the final phase $2\pi$ (figure 14 b4), the low-speed motion further intensifies, locating directly above the reference transducer. The corresponding strong ejection motion leads to the ZC $^+$ event ( ${\rm d}p'/{\rm d}t \gt 0$ ). One can find that the occurrence of pressure peaks and ZC events for IMF $^p_2$ is related to the convection of the hairpin-related vortices, highlighting the modulation on wall-pressure fluctuations by the vortical structures at different relative locations.

For higher-order modes of IMF $^p_3$ (figure 14 c), the phase evolution of turbulent structures with larger length approximately $300l$ remains similar to that of IMF $^p_2$ . The strength of low-speed structures increases to a level similar to the high-speed ones. The former structures correspond to the convection of high-speed regions at the spanwise side of the hairpin packets ( $u'\gt 0$ ) and low-speed regions ( $u'\lt 0$ ) within packets, respectively (Ganapathisubramani, Longmire & Marusic Reference Ganapathisubramani, Longmire and Marusic2003; Marusic & Monty Reference Marusic and Monty2019; Baars & Marusic Reference Baars and Marusic2020). For $\text{IMF}^p_4$ (figure 14 d), the structures turn to elongated streaks with larger length over $500l$ , exhibiting much weaker wall-normal motions. The turbulent structures responsible for the positive and negative pressure peaks change into the single upstream high- and low-speed motions, respectively (phases $\pi/2$ and $3\pi/2$ , figures 14 d1,d3). When the ZC events occur (phases $\pi$ and $2\pi$ , figures 14 d2,d4), a pair of velocity streaks emerges. The sign change of velocity streaks agrees with that of the wall-pressure fluctuations.