2.1. Signal processing

A summary of the signal processing adopted throughout the paper is presented here. The DNS volume data are appropriately scaled to obtain dimensional quantities. In particular, the numerical pressure data are rescaled by $\rho U^{2}_{\infty }$. The wall-pressure statistics are then computed for the four locations corresponding to sensors $7, 9, 21$ and $24$.

For all four sensor locations and even for those close to the TE in the strong APG region, the local pressure field has been found homogeneous in planes parallel to the wall and stationary in time, both numerically (Grasso et al. Reference Grasso, Jaiswal, Wu, Moreau and Roger2019) and experimentally (Jaiswal Reference Jaiswal2020; Jaiswal et al. Reference Jaiswal, Moreau, Avallone, Ragni and Pröbsting2020). In practical applications, only single time histories of the stochastic variable are available, and thus the stationary process is normally assumed to be ergodic. Under such conditions, the space–time cross-correlation function of the wall-pressure fluctuations $p'=p-\bar {p}$ at two arbitrary space–time points is defined as

(2.2)\begin{equation} R_{pp}(\boldsymbol{\xi}; \tau) = \lim_{T \rightarrow \infty} \frac{1}{T} \int^{T}_0 p'(\boldsymbol{x},t) p'(\boldsymbol{x} + \boldsymbol{\xi},t+\tau) \,\mathrm{d}t, \end{equation}

where $\boldsymbol {\xi }$ is the spatial separation vector between two points located at $\boldsymbol {x}$ and $\boldsymbol {x} + \boldsymbol {\xi }$ with time delay $\tau$. In this way, the random sample in space and time is compressed into a much shorter function of $\boldsymbol {\xi }$ and $\tau$. The cross-spectral density (CSD) is then calculated with a simple fast Fourier transformation using a Welch periodogram technique and Hanning windowing with zero padding (Salze et al. Reference Salze, Bailly, Marsden, Jondeau and Juvé2014),

(2.3)\begin{equation} \varPsi_{pp}(\boldsymbol{\xi};\omega) = \frac{1}{2{\rm \pi}} \int^{\infty}_{-\infty} R_{pp}(\boldsymbol{\xi}, \tau) \exp (-\mathrm{i}\omega \tau) \,\mathrm{d}\tau. \end{equation}

The spectra are computed using a Welch periodogram method with $16$ windows, each containing $512$ points, and with $50\,\%$ overlap. The high-frequency numerical noise is filtered out using a Butterworth filter. By doing a two-dimensional (2D) spatial Fourier transform of $\varPsi _{pp}(\boldsymbol {\xi };\omega )$ with the cross-correlation spectra for each time block averaged together, the CSD function in the wavenumber-frequency domain is then computed by discretising the following Fourier integral

(2.4)\begin{equation} \varPsi_{pp}(\boldsymbol{k};\omega) = \frac{1}{(2{\rm \pi})^2} \int\int^{\infty}_{-\infty} \varPsi_{pp}(\boldsymbol{\xi};\omega) \exp (-\mathrm{i}\boldsymbol{k} \boldsymbol{\xi})\, \mathrm{d}\boldsymbol{\xi}, \end{equation}

where $\boldsymbol {k}=(k_x,k_z)$ is the 2D wavevector (Bull Reference Bull1996), with $k_x$ and $k_z$ the streamwise and spanwise wavenumbers, respectively. The PSD or single-point wall-pressure spectrum (i.e. auto-spectrum), $\phi _{pp}$, at a given angular frequency $\omega$, is related to the space–time correlation function and the wavenumber-frequency spectrum by

(2.5)\begin{align} \phi_{pp}(\omega)& = \frac{1}{2{\rm \pi}} \int^{\infty}_{-\infty} R_{pp}(0;\tau) \exp (-\mathrm{i}\omega \tau) \,\mathrm{d}\tau,\nonumber\\ & = \int\int^{\infty}_{-\infty}\varPsi_{pp}(\boldsymbol{k};\omega)\,\mathrm{d}\boldsymbol{k}. \end{align}

The resolved normalised frequency range is $0\leq \omega \delta ^{*}/U_{\infty }\leq 5.53$ (where $\delta ^*$ is the boundary layer displacement thickness) with a frequency resolution of $\Delta \omega =1.4\times 10^{-3}U_{\infty }/\delta ^{*}$ and a wavenumber resolution of $\Delta k_x=0.12/\delta ^{*}$ in the streamwise direction and $\Delta k_z=0.14/\delta ^{*}$ in the spanwise direction.

4.1. PSD distribution

The PSD of the wall-pressure fluctuations at a given frequency, $\phi _{pp}(\,{f})$, is calculated from the wall-pressure CSD, as defined in § 2.1. The spanwise-averaged wall-pressure spectra in decibels with respect to the reference pressure $2\times 10^{5}$ Pa for a spectral resolution of 1 Hz (${\rm dB}\,{\rm Hz}^{-1}$ in short) are shown in figure 12 for the four locations on the aerofoil suction side indicated in figure 1. The PSD calculated from DNS are compared with the available experimental data from UdeS measurements (Jaiswal Reference Jaiswal2020; Jaiswal et al. Reference Jaiswal, Moreau, Avallone, Ragni and Pröbsting2020). Note that the experimental results for sensor $7$ are missing because of a microphone failure. In figures 12(a,b), for the ZPG regions $7$ and $9$, the semi-empirical spectrum defined by Goody (Reference Goody2004) is also represented. Such a model compares well with experimental data over a large range of Reynolds numbers (Hwang, Bonness & Hambric Reference Hwang, Bonness and Hambric2009), and is able to describe the essential properties of the single-point wall-pressure spectrum for ZPG boundary layers based on a limited numbers of variables. Note that at low frequencies, the same $-0.4$ slope is recovered as observed in the APG DNS of Na & Moin (Reference Na and Moin1998). For sensor $24$ (figure 12d) the DNS spectrum shows a small hump around 10 kHz. This hump in the higher-frequency range is also shown in the experimental data (grey line) and corresponds to an extra acoustic source in the wake (Wu et al. Reference Wu, Moreau and Sandberg2020). Going towards the TE, there is an increase in spectral levels of approximately $10\,{\rm dB}\,{\rm Hz}^{-1}$ at the low frequencies, and a faster roll-off decrease towards high frequency. This is an APG effect, also observed in previous studies (Na & Moin Reference Na and Moin1998; Cohen & Gloerfelt Reference Cohen and Gloerfelt2018). In the low-frequency range a plateau below $500$ Hz is observed for sensors $21$ and $24$, whereas sensors $7$ and $9$ show the characteristic ZPG quadratic rise with frequency following Goody's empirical model. Overall, the DNS and experimental results are in good agreement. The difference of PSDs between numerical and experimental results in the low-frequency range, approximately under $500$ Hz, is most likely related to installation effects, either attributed to jet noise (Moreau et al. Reference Moreau, Henner, Iaccarino, Wang and Roger2003) or to the unsteady interaction between the jet shear layer and the aerofoil, as well as the low background turbulence intensity (less than 0.4 %) observed in the experiments (Padois et al. Reference Padois, Laffay, Idier and Moreau2015) and not included in this DNS (Wu et al. Reference Wu, Moreau and Sandberg2020).

Figure 12. Spanwise-averaged wall-pressure spectra at four locations on the aerofoil suction side: ——, DNS-NS; $-\,{\cdot }\,-$, Goody's model, and —— (grey), UdeS measurements; (a) $x/c=-0.60$, (b) $x/c=-0.47$, (c) $x/c=-0.14$ and (d) $x/c=-0.08$.

In order to deduce the main contributors to the wall-pressure fluctuations in different frequency bands, the PSD and the frequency are then normalised based on several inner, outer and mixed scalings in figure 13. Different normalisations have been proposed over the years (Schloemer Reference Schloemer1967; Choi & Moin Reference Choi and Moin1990; Farabee & Casarella Reference Farabee and Casarella1991; Keith, Hurdis & Abraham Reference Keith, Hurdis and Abraham1992; Bull Reference Bull1996; Na & Moin Reference Na and Moin1998; Cipolla & Keith Reference Cipolla and Keith2000). There is no single universal scaling that leads to an acceptable collapse of data in the entire frequency range due to the multiscale nature of turbulence. However, as observed by Bull (Reference Bull1996) for ZPG flows, the wall-pressure fluctuations at low frequencies originate predominately from the outer layer, whereas those at high frequencies, where the spectrum varies as $\omega ^{-5}$ (Gravante et al. Reference Gravante, Naguib, Wark and Nagib1998; Na & Moin Reference Na and Moin1998; Moreau & Roger Reference Moreau and Roger2005), originate mostly from the buffer layer. Consequently, the wall-pressure PSD scales on outer variables at lower frequencies and inner variables at high frequencies. In the mid-frequency range, a scale-independent overlap region exists, where the PSD is proportional to $\omega ^{-1}$ (Bradshaw Reference Bradshaw1967; Gravante et al. Reference Gravante, Naguib, Wark and Nagib1998). This range is associated with motions in the logarithmic region.

Figure 13. Dimensionless spanwise-averaged wall-pressure spectra at four locations on the aerofoil suction side: (a) inner variables scaling, (b) mixed inner–outer variables scaling and (c,d) outer variables scaling. ZPG locations $x/c=-0.60$ (thin grey line) and $x/c=-0.47$ (thick grey line); APG locations $x/c=-0.14$ (thin black line), $x/c=-0.08$ (thick black line); $\bullet$, data by Choi & Moin (Reference Choi and Moin1990); $\circ$, ZPG data $x=0.5$ m, and $\square$, APG data $x=0.85$ m for attached TBL by Na & Moin (Reference Na and Moin1998); $\triangle$, ZPG (black), APGs (red) and APGw (magenta) data by Cohen & Gloerfelt (Reference Cohen and Gloerfelt2018).

The best inner variable scaling for the high-frequency range has been found by plotting $\phi _{pp}(\omega )u^{2}_{\tau }/\tau ^{2}_{w}\nu$ versus $\omega \nu /u^{2}_{\tau }$ and is used in figure 13(a). For the mid-frequency range, a satisfactory data collapse is usually obtained by scaling $\phi _{pp}(\omega )$ using the mixed inner–outer variables, $U_e$, $\delta ^{*}$ and $\tau _{w}$, with dimensionless frequency $\omega \delta ^{*}/U_e$ (Choi & Moin Reference Choi and Moin1990; Keith et al. Reference Keith, Hurdis and Abraham1992; Na & Moin Reference Na and Moin1998). This is shown in figure 13(b). Finally, in figure 13(c,d), common outer variable scalings for the low-frequency range are used: $\phi _{pp}(\omega ) U_e/p^{2}_{rms}\delta ^{*}$ and $\phi _{pp}(\omega ) U_e/q_{e}^{2}\delta ^{*}$, both versus $\omega \delta ^{*}/U_e$ (Choi & Moin Reference Choi and Moin1990; Na & Moin Reference Na and Moin1998). Here, $q_{e}$ is the dynamic pressure at the edge of the boundary layer.

First, the focus is given to the ZPG data sets. Normalisation with inner variables (figure 13a) collapses the spectra at high frequencies as expected, consistently with the findings of Na & Moin (Reference Na and Moin1998) and Choi & Moin (Reference Choi and Moin1990). Moreover, the present spectra at sensors $7$ and $9$ collapse well in the whole spectral range with that of Choi & Moin (Reference Choi and Moin1990), as the flow is characterised by a similar range of ${Re}_{\theta }$ (see table 1). In figure 13(b), the scaling with $U_e/\tau _{w}^{2} \delta ^*$ gives an overall collapse at low to mid frequencies, which was also observed by Na & Moin (Reference Na and Moin1998), Choi & Moin (Reference Choi and Moin1990) and Cohen & Gloerfelt (Reference Cohen and Gloerfelt2018). In addition, the effect of Reynolds numbers on the PSD is shown. As observed by Goody (Reference Goody2004), the main effect of Reynolds number on the wall-pressure spectrum is to widen its overlap range and shift the roll-off towards higher frequencies. This is indeed shown in figure 13(c,d), comparing data from Cohen & Gloerfelt (Reference Cohen and Gloerfelt2018) with those from Choi & Moin (Reference Choi and Moin1990) and those from sensor 7 to those from sensor 24 at a higher Reynolds number, for example. This is caused by the widening of the spectrum of turbulent motions and a larger logarithmic region associated with a higher-Reynolds-number boundary layer. Furthermore, a region of the spectra with a $-5$ slope in the high-frequency range, linked to the turbulent motions inside the buffer zone, is also shown in figure 13(c).

As the pressure gradient changes from zero to adverse (from sensor $9$ to sensor $24$), an overall increase in spectral levels normalised by the inner velocity scale is shown in figure 13, suggesting an augmentation of wall-pressure r.m.s. due to the APG, relative to $u_\tau$. In the presence of an APG the inner variables normalisation does not collapse the spectra at high frequencies, consistently with the observations of Na & Moin (Reference Na and Moin1998) and Cohen & Gloerfelt (Reference Cohen and Gloerfelt2018). This is because of the decrease in wall shear stress under APG. When the spectra are normalised by wall-pressure r.m.s. and $\delta ^*$ (figure 13c), a collapse in the low- to mid-frequency range is observed. This is expected because $p_{rms}^2$ represents the integral of the spectrum. The scaling with local dynamic pressure (figure 13d), on the other hand, does not give a good collapse in the low- to mid-frequency range, which is consistent with what Brooks & Hodgson (Reference Brooks and Hodgson1981) observed on the NACA0012 aerofoil (figure 11 in Brooks & Hodgson Reference Brooks and Hodgson1981). This is because the APG leads to augmentation of wall-pressure fluctuations at low frequencies, whereas it is associated with free-stream deceleration, i.e. a reduction in the edge dynamic pressure. Na & Moin (Reference Na and Moin1998) also observed similar comparison between ZPG and APG wall-pressure spectra for the normalisations analysed herein. Given that all cases plotted here differ in Reynolds numbers, pressure gradients and wall curvature, the spectra do not overlap over the entire frequency range. In addition, both ZPG and APG data of Cohen & Gloerfelt (Reference Cohen and Gloerfelt2018) display a faster roll-off at high frequencies, which is most likely a consequence of the spatial cut-off of the LES. These results also indicate that scalings used in popular semi-empirical models, especially using wall shear stress as the pressure scale, may be inappropriate for strong APG flows.

In figure 14, the pressure spectra are plotted again normalised using $\delta$ instead of $\delta ^{*}$, which is equivalent to using the Zaragola–Smits scaling instead of $U_{e}$ in figure 13 (because $\omega \,\delta /U_e=\omega \,\delta ^*/U_{ZS}$ and $U_e/\delta =U_{ZS}/\delta ^*$). ZPG and APG data of Cohen & Gloerfelt (Reference Cohen and Gloerfelt2018) for which all parameters are available are also shown. As shown by figure 6(b,c) and also indicated by Keith et al. (Reference Keith, Hurdis and Abraham1992) and Farabee & Casarella (Reference Farabee and Casarella1991), $\delta$ and $\delta ^{*}$ are not equivalent length scales given that their ratio varies with Reynolds number. For the outer scaling, both figure 14(a,b) show a significantly improved collapse compared with figure 13(c,d). This indicates that $\delta$ is a better length scale in the low- to mid-frequency range than $\delta ^*$, or equivalently that $U_{ZS}$ is a better velocity scale than $U_e$. Similarly, for the mixed inner–outer variable scaling, figure 14(c) shows a slightly better merge of the normalised spectra at mid-to-high frequencies for both ZPG and APG regions than in figure 13(b). Finally, the best overall collapse of normalised spectra in the whole frequency range is found in figure 14(a), which stresses that $p_{rms}$ is a better pressure scale than $q_{e}$ for the present highly non-equilibrium TBL. Hence, in development of wall-pressure spectra models, more focus is needed on modelling pressure gradient effects on $p_{rms}$ or on variables that scale with it.

Figure 14. Spanwise-averaged wall-pressure spectra at four locations on the aerofoil suction side scaled with Zaragola–Smits scaling, $U_{ZS}=U_{e}\delta ^{*}/\delta$: ZPG locations $x/c=-0.60$ (thin grey line) and $x/c=-0.47$ (thick grey line); APG locations $x/c=-0.14$ (thin black line), $x/c=-0.08$ (thick black line); $\triangle$, ZPG (black), APGs (red), APGw (magenta) data by Cohen & Gloerfelt (Reference Cohen and Gloerfelt2018). (a,b) Outer and (c) mixed inner–outer variables scaling.

In figure 15(a), the wall-pressure r.m.s. normalised by $\rho U^{2}_{in}$ (where $U_{in}$ is taken equal to the edge velocity at sensor $7$) as a function of wall-normal distance is compared with that of Na & Moin (Reference Na and Moin1998), for both attached and separated TBLs. Three different TBL cases by Na & Moin (Reference Na and Moin1998) are also considered, characterised by ZPG at $x/\delta ^*_{in}=80$, a weak APG at $x/\delta _{in}^{*}=120$ and a stronger APG just after the detachment at $x/\delta ^*_{in}=160$ therein. A local maximum of $p_{rms}$ is observed and both its elevation and magnitude increase as the flow moves downstream (with strengthened APG) in accordance with the trend shown by Na & Moin (Reference Na and Moin1998). Close to the TE (sensors 21 and 24) the levels and the hump size are close to those near the onset of separation in the flow studied by Na & Moin (Reference Na and Moin1998) ($x/\delta ^*_{in}=160$), highlighting the high load of the present case, consistently with the shape factor shown in figure 5(c) and in table 1. The streamwise increase of the pressure fluctuations at the wall is also shown in figure 15(b), which shows the streamwise distribution of the wall-pressure r.m.s. normalised by the local maximum magnitude of Reynolds shear stress or the local dynamic pressure. There is clearly less streamwise variation of the wall-pressure r.m.s. when normalised with local peak Reynolds shear stress, as also shown by Na & Moin (Reference Na and Moin1998). Thus, the peak Reynolds shear stress magnitude could be a better scaling variable for wall-pressure fluctuations than $U_e^2$ for all APG flows including the present strong non-equilibrium TBL case.

Figure 15. Root-mean-square values of (a) pressure fluctuations r.m.s. along wall-normal direction and (b) streamwise distribution of the wall-pressure r.m.s. normalised with local Reynolds shear stress ($\,{\cdot }\, {\cdot }\, {\cdot }\, $) or dynamic pressure ($-\,{\cdot }\,-$). ZPG locations $x/c=-0.60$ (thin grey line) and $x/c=-0.47$ (thick grey line); APG locations $x/c=-0.14$ (thin black line), $x/c=-0.08$ (thick black line). Na & Moin (Reference Na and Moin1998) data: $-\,{\cdot }\,-$, ZPG data at $x=0.5$ m, $--$, APG data at $x=0.85$ m for attached TBL (black); ——, $x/\delta ^*_{in}=80$, $--$ $x/\delta ^*_{in}=120$, $\,{\cdot }\,{\cdot }\,{\cdot }\, $, $x/\delta ^*_{in}=160$ for separated TBL (red).

In figures 16(a,b) the root-mean-square values of the wall-pressure scaled with $\tau _{w}$ or the dynamic pressure versus the friction Reynolds number, ${Re}_{\tau }=u_{\tau }\delta /\nu$, are shown at the four locations. In figure 16(c), the CD aerofoil distribution of $p^{+}_{rms}$ is also plotted versus the momentum-thickness-based Reynolds number. ZPG data of Choi & Moin (Reference Choi and Moin1990), DNS data of Abe et al. (Reference Abe, Matsuo and Kawamura2005) and ZPG data of Spalart (Reference Spalart1988), Cohen & Gloerfelt (Reference Cohen and Gloerfelt2018), Skote et al. (Reference Skote, Henningson and Henkes1998) and Schewe (Reference Schewe1983) are considered for comparison of ZPG data. The empirical law of $p^{+}_{rms}$ versus ${Re}_{\tau }$ developed for ZPG flows by Farabee & Casarella (Reference Farabee and Casarella1991) is also plotted in figure 16(a), in which $(\,p^{+}_{rms})^{2}= 6.5+1.86\ln ({Re}_{\tau }/333)$ for ${Re}_{\tau }>333$ and $(\,p^{+}_{rms})^{2}= 6.5$ for ${Re}_{\tau }\leq 333$. For APG flows, data of Cohen & Gloerfelt (Reference Cohen and Gloerfelt2018), Simpson et al. (Reference Simpson, Ghodbane and McGrath1987) and Burton (Reference Burton1973) are compared here. The data show that $p^{+}_{rms}$ weakly increases with ${Re}_\tau$ in ZPG flows following the empirical law proposed by Farabee & Casarella (Reference Farabee and Casarella1991). For APG flows, a greater scatter is observed, particularly in figure 16(a) (see APG locations $x/c=-0.14$ and $x/c=-0.08$ in the present flow and strong APG data of Burton (Reference Burton1973) and Simpson et al. (Reference Simpson, Ghodbane and McGrath1987)). APG leads to higher $p_{rms}^+$ than in a ZPG flow at the same Reynolds number. In comparison, less scatter is observed in figure 16(b), indicating a better correlation between the wall-pressure r.m.s. and the edge dynamic pressure than that with $\tau _w$. Finally, the sharp increase in $p_{rms}$ with $x$ shown in figure 16 for the present aerofoil further confirms the strong non-equilibrium state of the boundary layer at the TE, subject to a sharp APG.

Figure 16. Root-mean-square values of the wall-pressure r.m.s.: (a) $\tau _{w}$ scaling and (b) dynamic pressure scaling vs the friction Reynolds number; (c) $\tau _{w}$ scaling versus the momentum-thickness-based Reynolds number. ZPG locations $\circ$, $x/c=-0.60$ and $\square$, $x/c=-0.47$; APG locations $\triangledown$, $x/c=-0.14$ and $\boldsymbol {\pmb {+}}$, $x/c=-0.08$; $\bullet$, Choi & Moin (Reference Choi and Moin1990)'s data; Abe et al. (Reference Abe, Matsuo and Kawamura2005)'s data (${\star }$, grey, ${Re}_{\tau }=180$; $\Diamond$, ${Re}_{\tau }=395$; ${\Diamond }$, grey, ${Re}_{\tau }=640$); $\blacktriangleleft$, Spalart (Reference Spalart1988)'s data; $\bigstar$, ZPG (black), APGs (red) and APGw (magenta) data by Cohen & Gloerfelt (Reference Cohen and Gloerfelt2018); $\triangleleft$, Simpson et al. (Reference Simpson, Ghodbane and McGrath1987) APG's data; $\blacktriangle$, Burton (Reference Burton1973) APG's data; $\dots$, empirical law from Farabee & Casarella (Reference Farabee and Casarella1991); $\bullet$ (green), ZPG data by Skote et al. (Reference Skote, Henningson and Henkes1998) ${Re}_{\theta }=350-525$; $-\,{\cdot }\,-$ CD aerofoil distribution; $\blacktriangleleft$ (blue) data by Schewe (Reference Schewe1983).

In summary, the usual inner or inner–outer variable scalings are observed to work in the ZPG regions of the CD aerofoil (despite the possible history effects of the upstream FPG region), but not in the APG regions, with larger departures for higher $\beta _c$, as found in previous studies. The best scaling parameters are found to be the outer variables $U_e$, $\delta$ and $p_{rms}$ or equivalently $U_{ZS}$, $\delta ^*$ and $p_{rms}$. This indirectly stresses that the overall wall-pressure fluctuations are dominated by the large turbulent scales, the contribution of which increases with higher $\beta _c$. In turn, $p_{rms}$ is observed to scale more with outer variables such as the dynamic pressure $q_e$ than with inner scales such as $\tau _w$. As found previously by Na & Moin (Reference Na and Moin1998) and Abe (Reference Abe2017), for instance, a quasi invariant is obtained when scaling $p_{rms}$ with the local maximum Reynolds shear stress magnitude for the whole range of Reynolds number considered here in a strong non-equilibrium TBL typical of low-speed rotating machines. This has an important implication in choosing correct scalings in wall-pressure spectra models. For instance, most models use the wall shear stress as the pressure scale. However, our results show that $p_{rms}$ or the peak Reynolds shear stress magnitude are more appropriate low-frequency pressure scales. Another important finding is that, for low-Reynolds-number cases such as the current case, using $\delta$ and $U_e$ as length and velocity scales may give good overall collapse of wall-pressure spectra, as shown in figure 14(a).

4.2. Two-point correlations and coherence functions

The contour plots of the two-point correlation functions of wall-pressure fluctuations, introduced in § 2.1, are shown in figure 17 as a function of streamwise ($\xi _x$) and temporal ($\Delta t$) separations and normalised with inlet parameters. The slope ${\rm d}(\xi _x)/{\rm d}(\Delta t)$ gives the propagation speed of the eddies, the signatures of which are reflected in wall-pressure fluctuations. As a reference, these slopes of the cross-correlation at each spatial separation found in Na & Moin (Reference Na and Moin1998) flat-plate case of attached TBL, with ZPG at $x=0.50$ m and with APG at $x=0.85$ m, are also indicated by dotted-dashed and dashed red lines, respectively. As expected, a larger propagation speed is observed in the ZPG region, whereas in the APG region the convection speed of the turbulent structures is reduced and the contour plots are broader. This is associated with a longer correlation time of wall-pressure fluctuations in APG region, and it suggests a lower convection velocity of turbulent structures, which are discussed later. Variations of both the slope and broadness of the contours with APG are in agreement with the observations of Na & Moin (Reference Na and Moin1998), despite different ${Re}_{\theta }$ ranges and a stronger APG reached in the present flow.

Figure 17. Contour plot of two-point correlation of wall-pressure fluctuations versus streamwise spatial and temporal separations: (a) $x/c=-0.60$; (b) $x/c=-0.47$; (c) $x/c=-0.14$; and (d) $x/c=-0.08$. Contour levels are from 0.15 to 0.9 with increments of 0.15. Red lines are collections of outermost point of each iso-contour line for the attached TBL by Na & Moin (Reference Na and Moin1998) to indicate inclinations of the iso-contours: $-\,{\cdot }\,-$ (red), ZPG data at $x=0.5$ m, and $--$ (red), APG data at $x=0.85$ m.

By Fourier transforming the two-point correlation function, the CSD $\varPsi _{pp}(\boldsymbol {\xi };\omega )$ is calculated (see § 2.1). The cross-spectrum $\varPsi _{pp}(\boldsymbol {\xi };\omega )$ is useful for studying the wall-pressure spatial coherence as a function of frequency (Moreau & Roger Reference Moreau and Roger2005; Roger & Moreau Reference Roger and Moreau2005). In general, the coherence function between two points on the aerofoil surface can be defined as

(4.1)\begin{equation} \gamma^{2}(\boldsymbol{\xi};\omega)=\frac{|\varPsi_{pp}(\boldsymbol{\xi};\omega)|^{2}}{\phi_{pp}(\omega)^{2}}. \end{equation}

The coherence function describes the correlation between two points with a spatial separation of $\xi _x$ in the longitudinal (or streamwise) direction and $\xi _z$ in the lateral (or spanwise) direction at a given frequency. The contours of the calculated coherence distributions are shown along the streamwise and spanwise directions in figures 18 and 19, respectively. On the one hand, both streamwise and spanwise coherence functions are not observed to vary much in the ZPG regions, with a larger coherence in the streamwise direction. On the other hand, the streamwise coherence is observed to be reduced by APG for the whole frequency range, whereas the spanwise coherence overall increases with APG, especially at low frequencies, as already noted by Wang et al. (Reference Wang, Moreau, Iaccarino and Roger2009) based on the first LES on the present aerofoil. Figure 19 also indicates that the spanwise extent of the computational domain (i.e. 0.12 $C$) is sufficient for proper flow development in both ZPG and APG regions for most frequencies beyond 200 Hz, as already pointed out by Wang et al. (Reference Wang, Moreau, Iaccarino and Roger2009) and confirmed experimentally in several test facilities (Moreau & Roger Reference Moreau and Roger2005; Jaiswal Reference Jaiswal2020; Jaiswal et al. Reference Jaiswal, Moreau, Avallone, Ragni and Pröbsting2020). A slight increase of coherence level in the high-frequency range is found for sensor 24, which is caused by an acoustic contamination by an extra noise source in the wake, as already shown in the wall-pressure PSD in figures 12–14 and as mentioned by Wu et al. (Reference Wu, Moreau and Sandberg2020).

Figure 18. Contours of streamwise coherence of fluctuating pressure for ZPG and APG regions: (a) $x/c=-0.60$; (b) $x/c=-0.47$; (c) $x/c=-0.14$; and (d) $x/c=-0.08$. Symbols: $\cdots$ (red) spatial separations $\xi _x/c=0.007, 0.013 ,0.016$ used in figures 20 and 21; $-\,{\cdot }\,-$ (blue) frequencies of 1600 and 5000 Hz used in figure 22 and 23.

Figure 19. Contours of spanwise coherence of fluctuating pressure for ZPG and APG regions: (a) $x/c=-0.60$; (b) $x/c=-0.47$; (c) $x/c=-0.14$; and (d) $x/c=-0.08$. Symbols: $\cdots$ (red), spatial separations $\xi _z/c=0.0019, 0.0025, 0.0049$ used in figures 20 and 21; $-\,{\cdot }\,-$ (blue) frequencies 1600 and 5000 Hz used in figures 22 and 23.

More quantitative estimates are provided by the magnitudes of the normalised longitudinal and lateral CSD, $\gamma (\xi _x,\omega )= |\varPsi _{pp}(\xi _x, 0, \omega )| / \phi _{pp}(\omega )$ and $\gamma (\xi _z,\omega )=|\varPsi _{pp}( 0,\xi _z, \omega )|/\phi _{pp}(\omega )$, respectively. First, figures 20 and 21 show these normalised CSD versus the normalised frequency $\omega \delta _{in}^{*} / U_{\infty }$ for different spatial separations, indicated with horizontal red dotted lines in figures 18 and 19. In figure 20, the normalised longitudinal CSD in APG is shown to be systematically below that for a ZPG, as already noted by Schloemer (Reference Schloemer1967). The levels are higher at sensor 7 than at sensor 9, most likely a history effect of the upstream FPG flow. The decay of coherence with frequency is also shown to be faster in APG (with a power-law slope of $-0.4$) than in ZPG (with a slope of $-0.32$). In figure 20, $\gamma (\xi _x,\omega )$ is also compared with distributions provided by Hu (Reference Hu2021), who measured wall-pressure fluctuations induced by TBL with different pressure gradients by installing a NACA0012 aerofoil above a flat plate with an adjustable angle of attack. Even though the Reynolds numbers ${Re}_\theta$ are much larger in the latter experiments, good agreement is found in the variation of streamwise coherence with frequencies for both ZPG and APG conditions. Only at high frequencies, a slightly faster decay in ZPG is observed in the distributions of Hu (Reference Hu2021), but the correlations are low and bear the largest experimental uncertainties. Such a good match underlines that the streamwise wall-pressure coherence is quite insensitive to a significant variation in ${Re}_\theta$, similar to what was observed in the velocity correlations by Sillero, Jiménez & Moser (Reference Sillero, Jiménez and Moser2014) in ZPG TBL on flat plates and by Pargal et al. (Reference Pargal, Wu, Yuan and Moreau2022) with the present DNS data of a CD aerofoil and with another DNS of flow on a flat plate with the same external mean pressure gradient. In figure 21, the normalised lateral CSD are shown to vary far less with flow conditions than the longitudinal versions. Only at low to mid frequencies is a systematic increase of spanwise coherence observed with APG, consistently with the observations of Wang et al. (Reference Wang, Moreau, Iaccarino and Roger2009). This coherence increase in a given frequency range is predominantly associated with large turbulent structures formed during APG.

Figure 20. Streamwise $\gamma (\xi _x,\omega )$ frequency distribution for different spatial separations at four location on the aerofoil suction side: (a) $\xi _x/c = 0.007$ and $\xi _x/\delta _{in}^{*}=2.61$; (b) $\xi _x/c = 0.013$ and $\xi _x/\delta _{in}^{*}=5.19$; (c) $\xi _x/c = 0.016$ and $\xi _x/\delta _{in}^{*}=6.23$. ZPG locations $x/c=-0.60$ (thin grey line) and $x/c=-0.47$ (thick grey line); APG locations $x/c=-0.14$ (thin black line), $x/c=-0.08$ (thick black line). Symbols: $\cdots$ ZPG data by Hu (Reference Hu2021) (blue for $\xi _x/\delta ^{*}=4.3$, red for $\xi _x/\delta ^{*}=6.0$ and turquoise for $\xi _x/\delta ^{*}=6.3$); $-\,{\cdot }\,-$ APG data by Hu (Reference Hu2021) (blue for $\xi _x/\delta ^{*}=2.0$ and red for $\xi _x/\delta ^{*}=2.7$).

Figure 21. Spanwise $\gamma (\xi _z,\omega )$ frequency distribution for different spatial separations at four location on the aerofoil suction side: (a) $\xi _z/c = 0.0019$; (b) $\xi _z/c = 0.0025$; (c) $\xi _z/c = 0.0049$. ZPG locations $x/c=-0.60$ (thin grey line) and $x/c=-0.47$ (thick grey line); APG locations $x/c=-0.14$ (thin black line), $x/c=-0.08$ (thick black line).

Next, figures 22 and 23 show the coherence, $\gamma ^2$, versus various streamwise and spanwise separations at two increasing frequencies, indicated with vertical blue dashed-dotted lines in figures 18 and 19, for the four locations on the aerofoil. A faster decay of coherence with distance (in both $x$ and $z$) is observed for higher frequencies, regardless of the mean pressure gradient. The streamwise coherence in figure 22(a) decays rapidly with longitudinal distance when moving downstream, indicating shorter coherence in the APG region than in the ZPG region. However, the spanwise coherence decay with lateral distance in figure 23(b) does not show significant difference between the four chordwise locations. As also observed in Schloemer (Reference Schloemer1967), the spanwise coherence decays faster with distance than the streamwise coherence in both ZPG and APG regions for both frequencies. A similar trend was also found by Brooks & Hodgson (Reference Brooks and Hodgson1981) on the NACA0012 aerofoil (figure 16 in Brooks & Hodgson Reference Brooks and Hodgson1981).

Figure 22. Coherence functions, $\gamma ^{2}$, at various streamwise separations on the aerofoil suction side: (a) $1600$ Hz and (b) $5000$ Hz. ZPG locations $x/c=-0.60$ (thin grey line) and $x/c=-0.47$ (thick grey line); APG locations $x/c=-0.14$ (thin black line) and $x/c=-0.08$ (thick black line).

Figure 23. Coherence functions, $\gamma ^{2}$, at various spanwise separations on the aerofoil suction side: (a) $1600$ Hz and (b) $5000$ Hz. ZPG locations $x/c=-0.60$ (thin grey line) and $x/c=-0.47$ (thick grey line); APG locations $x/c=-0.14$ (thin black line) and $x/c=-0.08$ (thick black line).

All of this is also consistent with what has been observed previously for the main contributors to wall-pressure fluctuations in a TBL: on the one hand, the wall-pressure behaviour at high frequencies is predominantly controlled by smaller structures in the inner layer, with short delay times and steep coherence decay rates with distance (Schloemer Reference Schloemer1967; Van Blitterswyk & Rocha Reference Van Blitterswyk and Rocha2017); on the other hand, the bigger outer-layer structures, characterised by longer decay lengths, dominate the wall-pressure coherence at low to mid frequencies (Bull Reference Bull1996; Palumbo Reference Palumbo2012). The observed increase of spanwise coherence with APG in the low- to mid-frequency range can be related to the larger lifted turbulent structures shown in figure 11(b).

In figures 24 and 25, the magnitude of the normalised longitudinal and lateral CSD are plotted vs the phase $\omega \xi _{x,z}/U_c$, with $U_c$ the convective velocity, for several spatial separations for ZPG and APG cases. The streamwise $\gamma (\xi _x,\omega )$ and spanwise $\gamma (\xi _z,\omega )$ measure the loss of coherence of the eddies as they move downstream and, similarly, the decrease of coherence of the eddies in the spanwise direction, respectively (Schloemer Reference Schloemer1967). In figures 24 and 25, the dashed-dotted lines represent the curves found to fit the data of Corcos (Reference Corcos1964) for ZPG TBL on a flat plate, $\exp (-\alpha _x\omega \xi _x/U_c)$ with $\alpha _x=0.11$ and $\exp (-\alpha _z\omega \xi _z/U_c)$ with $\alpha _z=0.8$ and $0.714$. The convection velocities considered here correspond to about $U_c\approx 0.8U_{\infty }$ for ZPG flow and $U_c\approx 0.5U_{\infty }$ for APG flow, similarly to what was found by Na & Moin (Reference Na and Moin1998), who observed a mean convection velocity of $U_c\approx 0.79U_{\infty }$ in ZPG flow and $U_c\approx 0.56U_{\infty }$ in APG flow, or by Schloemer (Reference Schloemer1967) who found a mean convection velocity of $U_c\approx 0.75U_{\infty }$ in ZPG flow and $U_c\approx 0.54U_{\infty }$ in APG flow. The decay rates $\alpha _{x}$ and $\alpha _{z}$ at the four locations analysed here are obtained by fitting the data at different streamwise spatial separations and are plotted in the figures for each location using black solid lines. As shown in figure 24, the APG results display lower values of $\gamma (\xi _x,\omega )$ for a fixed $\omega \xi _{x}/U_c$ compared with the ZPG results, which corresponds to an increase of $\alpha _x$ in the APG region. Indeed, a good fit to the data is found for the exponential decay of $\alpha _x=0.11$ and $\alpha _x=0.14$ for the ZPG locations and $\alpha _x=0.23$ and $\alpha _x=0.29$ for the APG locations. For the present CD aerofoil, the ZPG coherence decays with the phase similarly (sensor 7) or more rapidly (sensor 9) than what was found by Corcos (Reference Corcos1964). These trends in both ZPG and APG are consistent with the findings of Hu (Reference Hu2021), Brooks & Hodgson (Reference Brooks and Hodgson1981) and Schloemer (Reference Schloemer1967). For instance, Hu (Reference Hu2021) reported $\alpha _x=0.14$ in ZPG flow and $\alpha _x=0.2$ in APG flow. In a flow with milder APG, Brooks & Hodgson (Reference Brooks and Hodgson1981) observed an $\alpha _x=0.19$. Given the difference in flow condition, this again suggests that the streamwise decay coefficient $\alpha _x$ hardly depends on the Reynolds number, consistently with the results of Sillero et al. (Reference Sillero, Jiménez and Moser2014) and Pargal et al. (Reference Pargal, Wu, Yuan and Moreau2022) on velocity correlations. In turn, this suggests that the empirical correlation of $\alpha _x$ with friction velocity or Reynolds number based on momentum thickness (equations (2) and (4) in Hu Reference Hu2021) may not apply at lower ${Re}_\theta$. In figure 24, the curves for different streamwise separations tend to collapse for frequencies higher than a characteristic value, which varies with sensor location, decreasing as it moves downstream. The collapse implies a constant decay rate of the wall-pressure fluctuations for the high-frequency range. These results are consistent with the comparisons shown in figures 18–22.

Figure 24. Streamwise $\gamma (\xi _x,\omega )$ vs the phase $\omega \xi _x/U_c$ for ZPG and APG positions: (a) $x/c=-0.60$; (b) $x/c=-0.47$; (c) $x/c=-0.14$; and (d) $x/c=-0.08$. Symbols — (blue), $\xi _x/c=0.013$; — (purple), $\xi _x/c=0.016$; – – (turquoise), $\xi _x/c=0.021$; $\cdots$ (red), $\xi _x/c=0.026$; $\cdots$ (green), $\xi _x/c=0.037$; $\cdots$ (olive), $\xi _x/c=0.061$; $-\,{\cdot }\,-$ (red), $\exp (-0.11\omega \xi _x/U_c)$; $-$, $\exp (-\alpha _x\omega \xi _x/U_c)$ with $\alpha _x=0.11$ for (a), $\alpha _x=0.14$ for (b), $\alpha _x=0.23$ for (c) and $\alpha _x=0.29$ for (d).

Figure 25. Spanwise $\gamma (\xi _z,\omega )$ versus the phase $\omega \xi _z/U_c$: (a) $x/c=-0.60$; (b) $x/c=-0.47$; (c) $x/c=-0.14$; and (d) $x/c=-0.08$. Symbols: — (blue), $\xi _z/c=0.0019$; — (purple), $\xi _z/c=0.0025$; – – (turquoise), $\xi _z/c=0.0031$; $\cdots$ (red), $\xi _z/c=0.0037$; $\cdots$ (green), $\xi _z/c=0.0043$; $\cdots$ (olive), $\xi _z/c=0.0056$; $-\,{\cdot }\,-$ $\exp (-0.714\omega \xi _z/U_c)$ and (red) $\exp (-0.8\omega \xi _z/U_c)$; —, $\exp (-\alpha _z\omega \xi _z/U_c)$ with $\alpha _z=0.76$ for (a), $\alpha _z=0.70$ for (b), $\alpha _z=0.48$ for (c) and $\alpha _z=0.55$ for (d).

On the other hand, in figure 25, similar smaller changes in $\gamma$ in the spanwise direction between ZPG and APG flows are found as observed by Schloemer (Reference Schloemer1967), Brooks & Hodgson (Reference Brooks and Hodgson1981) and Hu (Reference Hu2021). In figure 25, a good fit to the data is found for the exponential decay with $\alpha _z=0.76$ and $\alpha _z=0.70$ for the two locations in the ZPG region, respectively, and $\alpha _z=0.48$ and $\alpha _z=0.55$ for the APG regions, which yields a 36 % variation in the spanwise direction compared with about a factor of two in the streamwise direction. For instance, Hu (Reference Hu2021) reported $\alpha _z=0.75$ in ZPG and $\alpha _z=0.5$ for the APG TBL in table 1, which yields a similar 33 % variation in the spanwise direction. This, in turn, suggests that the spanwise decay coefficient $\alpha _z$ is weakly dependent on the Reynolds number, consistently with the results of Sillero et al. (Reference Sillero, Jiménez and Moser2014) and Pargal et al. (Reference Pargal, Wu, Yuan and Moreau2022) on velocity correlations. The variations of $\alpha _z$ in ZPG and APG flows are also consistent with the data shown in figures 19–23. The observed similarity in the decay coefficient variations across a wide range of Reynolds numbers can help modelling the spatial coherence of wall-pressure fluctuations, as an essential part of TE noise prediction.

In the present compressible DNS case, the range of frequency considered is $0.069\leq \omega \delta ^{*}/U_{\infty } \leq 1.12$ to neglect the high-frequency acoustic contribution (i.e. additional noise source in the wake), which produces an increase of the spanwise $\gamma (\xi _z,\omega )$ and streamwise $\gamma (\xi _x,\omega )$ distributions as indicated in figure 25(c) for $\xi _z/c=0.0043$ (green dotted line). In addition, for a given pressure gradient, the decay rates are quite sensitive to a change in the convection velocity. Specifically, a lower $U_c$ shifts $\gamma (\xi _{x,z},\omega )$ towards higher $\omega \xi _{x,z}/U_c$, which gives a smaller decay rate.

A coherence length of wall-pressure fluctuations at each frequency can then be defined by the following expression in both streamwise and spanwise directions:

(4.2)\begin{equation} L_{x,z} (\,{f}) = \int^{\infty}_{0}\gamma(\xi_{x,z},f) \,\mathrm{d}\xi_{x,z}. \end{equation}

The coherence length distributions in the same frequency range as previously are plotted in figure 26 for the four positions on the CD aerofoil. The measurements of Hu (Reference Hu2021) over a more limited frequency range for the ZPG and APG flow conditions in table 1 are also shown. Despite the different ${Re}_\theta$, the two sets of data show very similar results in the ZPG region for both coherence lengths. In the APG the trends are similar with smaller streamwise coherence length and larger spanwise length than in ZPG, but their roll-off occurs at a higher reduced frequency $\omega \,\delta ^*/U_e$ closer to 1 (Reynolds number effect similar to what is observed for the PSD of the wall-pressure fluctuations). Consequently both $L_z$ for ZPG and APG merge almost a decade later in frequency in the data of Hu (Reference Hu2021). The effect of APG on the coherence lengths is, in fact, consistent with what was found in Hu (Reference Hu2021) and Van Blitterswyk & Rocha (Reference Van Blitterswyk and Rocha2017). The difference in the roll-off between APG and ZPG profiles, for both $L_{x}(\,{f})$ and $L_{z}(\,{f})$, reflects the effect of APG on the PSD spectra (see figure 12). At low frequencies, both coherence lengths $L_{x,z}(\,{f})$ increase with APG significantly. At middle to high frequencies, lower levels are found for the APG profiles compared with the ZPG profiles, with a drop in the streamwise coherence length $L_x$ much more marked than in the spanwise length $L_z$ that stays almost similar at all locations. Indeed, as also observed by Van Blitterswyk & Rocha (Reference Van Blitterswyk and Rocha2017) at a higher ${Re}_{\theta }$, the intensity of smaller-scale structures with faster decay increases, as well as the size of large-scale structures.

Figure 26. Normalised coherence lengths for ZPG locations ($\circ$, $x/c=-0.60$, and $\square$, $x/c=-0.47$) and APG locations ($\triangledown$, $x/c=-0.14$, and $\boldsymbol {\pmb {+}}$, $x/c=-0.08$) in (a) streamwise and (b) spanwise directions: $\bullet$ (blue), ZPG data by Hu (Reference Hu2021) at ${Re}_{\theta }=4889$; $\blacktriangle$ (red), APG data by Hu (Reference Hu2021) at ${Re}_{\theta }=8670$.

Comparison between the streamwise correlation lengths normalised using inner (figure 27a) and outer (figure 27b) variables shows that a better collapse at all four locations is obtained with an inner scaling. However, for the spanwise correlation length, a better collapse is obtained using the outer variables (figure 28b compared with figure 28a).

Figure 27. Normalised streamwise coherence lengths for ZPG locations ($\circ$, $x/c=-0.60$, and $\square$, $x/c=-0.47$) and APG locations ($\triangledown$, $x/c=-0.14$, and $\boldsymbol {\pmb {+}}$, $x/c=-0.08$) in (a) inner and (b) outer variable scalings.

Figure 28. Normalised spanwise coherence lengths for ZPG locations ($\circ$, $x/c=-0.60$, and $\square$, $x/c=-0.47$) and APG locations ($\triangledown$, $x/c=-0.14$, and $\boldsymbol {\pmb {+}}$, $x/c=-0.08$) in (a) inner and (b) outer variable scalings.

4.3. Wavenumber-frequency spectra

The wavenumber-frequency spectral density of the wall-pressure fluctuations, $\varPsi _{pp}(\boldsymbol {k};\omega )$, is related to the CSD $\varPsi _{pp}(\boldsymbol {\xi };\omega )$ by a 2D spatial Fourier transform (see § 2.1). In figures 29 and 30, the normalised 2D wavenumber-frequency spectra, $\overline {\varPsi }_{pp}(\boldsymbol {k};\omega )= \varPsi _{pp}(\boldsymbol {k};\omega )(U_{\infty }/\tau _{w}^2\delta _{in}^{*3})$, is plotted in the normalised wavenumber domain $(k_x \delta _{in}^{*}, k_z\delta _{in}^{*})$ at three dimensionless frequency values, with $k_x$ and $k_z$ the streamwise and spanwise wavenumbers, respectively. Note that the same dimensionless frequencies are used for the four locations: a low frequency $\omega \delta _{in}^{*}/U_{\infty }=0.34$, a middle frequency $\omega \delta _{in}^{*}/U_{\infty }=0.76$ and a high frequency $\omega \delta _{in}^{*}/U_{\infty }=1.73$. At the low frequency, the convective contribution centred around the convective wavenumber $k_x=k_c$ (with $k_c=\omega /U_c$) is predominant. On the other hand, the acoustic contribution centred around the acoustic wavenumber $k_x=k_0$ is not visible. At the high frequency $\omega \delta _{in}^{*}/U_{\infty }=1.73$, however, the acoustic contribution appears and separates from the convective contribution, as shown in figure 30. The trace of the acoustic domain matches an ellipse with its centre at $(k_0 M/\beta ^{2},0)$, a major radius of $k_0/\beta ^{2}$ and a minor radius of $k_0/\beta$ with the Prandtl–Glauert factor given by $\beta =\sqrt {1-M^{2}}$ (see dashed red lines in figures 29 and 30), as also found in the compressible LES of Cohen & Gloerfelt (Reference Cohen and Gloerfelt2018). At the low frequency, the red ellipse merges with the convective domain, whereas at the higher frequencies it is clearly visible. As noted by Cohen & Gloerfelt (Reference Cohen and Gloerfelt2018), it is rare to capture the acoustic contribution and this is a first case in such a non-equilibrium TBL with high APG. At higher frequencies (${\approx }10$ kHz for APG locations as shown in figure 30), the acoustic contribution to the fluctuations becomes significant. Indeed, for higher frequencies, darker contours around the acoustic wavenumber and lighter contours in the convective region are shown in figures 29 and 30. The convective contribution shown in the figures has an antisymmetric distribution along $k_x$ and an elongated elliptical shape with a major axis in the $k_z$ direction. The convective ridge in $k_x$ in the ZPG spectra in figure 29 is slightly smaller (with higher $U_c$) than in the APG spectra in figure 30. This behaviour is consistent with what was found by Cohen & Gloerfelt (Reference Cohen and Gloerfelt2018), Salze et al. (Reference Salze, Bailly, Marsden, Jondeau and Juvé2015) and Bull (Reference Bull1996). Thus, as shown previously, the wall-pressure field is more coherent in the streamwise direction for the ZPG region than for the APG region (see § 4.2).

Figure 29. Normalised 2D wavenumber-frequency spectra $\overline {\varPsi }_{pp}(\boldsymbol {k};\omega )= \varPsi _{pp}(\boldsymbol {k};\omega )(U_{\infty }/\tau _{w}^2\delta _{in}^{*3})$ measured at three dimensionless frequencies: (a,d) $\omega \delta _{in}^{*}/U_{\infty }=0.34$; (b,e) $\omega \delta _{in}^{*}/U_{\infty }=0.76$; (c,f) $\omega \delta _{in}^{*}/U_{\infty }=1.73$. ZPG locations: (a–c) $x/c=-0.60$; (d–f) $x/c=-0.47$. Trace of the acoustic domain (red dashed line) (Cohen & Gloerfelt Reference Cohen and Gloerfelt2018) and centre of the convective ridge (red dotted line).

Figure 30. Normalised 2D wavenumber-frequency spectra $\overline {\varPsi }_{pp}(\boldsymbol {k};\omega )=\varPsi _{pp}(\boldsymbol {k};\omega )(U_{\infty }/\tau _{w}^2\delta _{in}^{*3})$ measured at three dimensionless frequencies: (a,d) $\omega \delta _{in}^{*}/U_{\infty }=0.34$; (b,e) $\omega \delta _{in}^{*}/U_{\infty }=0.76$; (c,f) $\omega \delta _{in}^{*}/U_{\infty }=1.73$. APG locations: (a–c) $x/c=-0.14$; (d–f) $x/c=-0.08$. Trace of the acoustic domain (red dashed line) (Cohen & Gloerfelt Reference Cohen and Gloerfelt2018).

In figure 31, the normalised wavenumber-frequency spectra for the four locations are also plotted as functions of frequency for increasing $k_x\delta _{in}^{*}$ in the range $0.36\leq k_x\delta _{in}^{*} \leq 2.11$, with increments of $0.25$ going from top (solid black line) to the bottom (solid thick grey line) of the plot. The maximum of each spectrum in figure 31 corresponds to the primary convective peak. For higher $k_x\delta _{in}^{*}$ (i.e. solid thick grey line with $k_x\delta _{in}^{*}=2.11$ in figure 31), the primary convective peak moves to higher frequencies (Choi & Moin Reference Choi and Moin1990; Bull Reference Bull1996). In the same way, as indicated in the contour plots of figures 29 and 30, as the frequency increases the convective ridge moves to higher wavenumbers. This behaviour can be analysed in more detail if considering, for example, figure 29(a) for $x/c=-0.60$; in this case, the convective ridge is centred around $k_x\delta _{in}^{*}= 0.36$ for $\omega \delta _{in}^{*}/U_{\infty }=0.34$. Thus, at this normalised frequency, in figure 31(a), a maximum is shown for the solid black curve which corresponds to $k_x\delta _{in}^{*}= 0.36$ (see the red dotted line in figures 29a and 31a). For APG, the acoustic peak is also observed at higher frequencies. If considering, for example, figure 29(f), for $x/c=-0.08$, the acoustic domain is centred around $k_x\delta _{in}^{*}= 0.40$ for $\omega \delta _{in}^{*}/U_{\infty }=1.73$. Thus, around this frequency in figure 31(d), a maximum is shown for the solid black curve which corresponds to a closer $k_x\delta _{in}^{*}$ of $0.36$. The result shows that the high-frequency humps observed in wall-pressure PSD in a non-equilibrium flow such as the present case are due to acoustic contribution. Therefore, it may not be appropriate to model high-frequency humps of this kind directly based on aerodynamic boundary layer parameters (e.g. by Dominique et al. Reference Dominique, Van den Berghe, Schram and Mendez2022).

Figure 31. Normalised 2D wavenumber-frequency spectra as function of frequency at increasing $k_x\delta _{in}^{*}$. $k_x\delta _{in}^{*}=0.36$ (solid black line) to $k_x\delta _{in}^{*}=2.11$ (solid thick grey line), with increments of $0.25$: (a) $x/c=-0.60$; (b) $x/c=-0.47$; (c) $x/c=-0.14$; and (d) $x/c=-0.08$.

4.4. Convection velocity

In this section, the convection velocity $U_c$ is calculated from the DNS database for the four locations under analysis. Generally speaking, the convection velocity of a Fourier mode is expressed as $U_c = \omega /k_x=f\lambda _x$. However, for a given frequency a spectrum of wavenumber is found, and vice versa, as shown in figures 29, 30 and 31 (Del Álamo & Jiménez Reference Del Álamo and Jiménez2009). In general, Taylor's frozen turbulent approximation (Taylor Reference Taylor1938) that assumes a uniform $U_c$ equal to the local mean velocity is commonly used. Such an approximation, acceptable for low frequencies and low wavenumbers, neglects the dependence of $U_c$ on eddy size leading to errors in the interpretation of large-scale structures in jets for instance (Del Álamo & Jiménez Reference Del Álamo and Jiménez2009). To overcome this limitation, different definitions of $U_c$ have been proposed, which consider its dependence on wavenumber, frequency and spatial separation (Wills Reference Wills1964; Goldschmidt, Young & Ott Reference Goldschmidt, Young and Ott1981; Hussain & Clark Reference Hussain and Clark1981).

In the following section, the convection velocities as functions of frequency, wavenumber or spatial separation are calculated for ZPG and APG TBLs on the CD aerofoil and compared with available data by Choi & Moin (Reference Choi and Moin1990), Na & Moin (Reference Na and Moin1998), Cohen & Gloerfelt (Reference Cohen and Gloerfelt2018) and Burton (Reference Burton1973).

4.4.1. Convection velocity as a function of frequency

The convection velocity $U_c(\omega )$ is extracted from a linear interpolation of the phase $\omega \xi _x/ U_c$ in the cross-spectrum, $\varPsi _{pp}(\boldsymbol {\xi };\omega )$, and scaled with inner and outer boundary layer variables, as shown in figure 32. The range of spatial separations $\xi _x$ for which the phase is linear within approximately $10\,\%$ is considered for the extraction of $U_c(\omega )$.

Figure 32. Normalised convection velocities as function of dimensionless frequency at four locations on the aerofoil suction side: ZPG locations $x/c=-0.60$ (thin grey line) and $x/c=-0.47$ (thick grey line); APG locations $x/c=-0.14$ (thin black line) and $x/c=-0.08$ (thick black line); $\bullet$, data by Choi & Moin (Reference Choi and Moin1990); (a) inner variables scaling; (b) outer variables scaling.

The inner variables scaling shown in figure 32(a) shows a good collapse of the spectra. In general, the convection velocity increases with frequency, reaching a maximum and then slightly decreases with frequency, as the size of structures in the boundary layer, proportional to the wavelength, decreases (Schloemer Reference Schloemer1967). For a given value of $\omega \delta /u_{\tau }$, the ratio $U_c(\omega )/U_{\infty }$ is lower for APG than for ZPG, because the smallest turbulent structures located near the wall are subjected to a slower flow, and the largest turbulent structures lifted from the wall mostly convect with the bulk velocity that is also lower (see figure 11). Therefore, smaller mean convection velocities of the turbulent eddies are found. In figure 32, as the frequency increases, $U_c(\omega )/U_{\infty }$ reaches approximately $0.8$ for the ZPG cases and $0.5$ for the APG cases, as previously found by Bull (Reference Bull1996) and used in § 4.2. Moreover, in APG the high-frequency limit slightly decreases. On the other hand, for ZPG the ratio $U_c(\omega )/U_{\infty }$ increases to a maximum asymptotic value. The ratio found for the ZPG regions, at sensor 9 noticeably, is in good agreement with Choi's ZPG data (Choi & Moin Reference Choi and Moin1990). At lower frequencies, the ratio decreases. This behaviour, also observed in Choi's ZPG data, is linked to the loss of coherence of the turbulent structures described in § 4.2 (figure 24). The convection velocity is slightly higher at sensor 7 most likely because of the history effect of the upstream FPG region (figure 15c in Schloemer Reference Schloemer1967). The mean convection velocities calculated from these distributions are $U_c\approx 0.87\,U_{\infty }$ for sensor $7$, $U_c\approx 0.80\,U_{\infty }$ for sensor $9$, $U_c\approx 0.50\,U_{\infty }$ for sensor $21$ and $U_c\approx 0.44\,U_{\infty }$ for sensor $24$. In turn, these ratios correspond to coherence decay rates $\alpha _x=0.13$ and $\alpha _z=0.82$ at sensor $7$, and $\alpha _x=0.25$ and $\alpha _z=0.49$ at sensor $24$, whereas at sensors $9$ and $21$ the coherence decay rates are as indicated in figures 24 and 25.

Overall, figure 32 shows uniform distributions for $U_c(\omega )$, consistent with previous measurements in both ZPG (see, for instance, figure 15a in Schloemer (Reference Schloemer1967)) and APG flows (see, for instance, figure 15b in Schloemer (Reference Schloemer1967) and figure 19 in Brooks & Hodgson (Reference Brooks and Hodgson1981)). The results also indicate that the convection velocity scales well with friction velocity for the majority of the frequency range, even in highly non-equilibrium flows. The differences observed at sensor 24 at low frequencies could be due to the vicinity of this location to the TE (Messiter Reference Messiter1970; Wu et al. Reference Wu, Moreau and Sandberg2020). Note also that the data presented do not extend to lower frequencies where the ratio may be sensitive to the sampling period.

4.4.2. Convection velocity as a function of streamwise wavenumber

The convection velocity can be defined from the wavenumber-frequency spectra, $\varPsi _{pp}(k_x,\omega )$, as function of the streamwise wavenumber by considering the maximum of the frequency spectrum at a given wavenumber $[\partial \varPsi _{pp}(k_x,\omega )/\partial \omega ]_{\omega =\omega _{c}(k_x)}=0$. Such a maximum corresponds to the convective frequency that gives $U_c(k_x)=-\omega _c(k_x)/k_x$. This scheme was proposed by Wills (Reference Wills1964) to consider the dependence of $U_c$ on both wavenumber and frequency, overcoming the artifact of Taylor's approximation that is most likely not verified in the present highly sheared strongly non-equilibrium TBL (Moin Reference Moin2009). Similarly, $U_c(k_x)$ can be defined considering the maximum of the wavenumber spectrum at a given frequency (Comte-Bellot & Corrsin Reference Comte-Bellot and Corrsin1971). The normalised convection velocities as functions of dimensionless streamwise wavenumber at the four locations on the aerofoil suction side are plotted in figure 33.

Figure 33. Normalised convection velocities as function of dimensionless streamwise wavenumber at four locations on the aerofoil suction side: ZPG locations $x/c=-0.60$ (thin grey line) and $x/c=-0.47$ (thick grey line); APG locations $x/c=-0.14$ (thin black line), $x/c=-0.08$ (thick black line); $\bullet$, data by Choi & Moin (Reference Choi and Moin1990) and $-\,{\cdot }\,-$, the model of Panton & Linebarger (Reference Panton and Linebarger1974) for ${Re}_{\tau }=178$ and ${Re}_{\tau }=185$, corresponding to sensors 7 (thin grey line) and 9 (thick grey line), respectively; (a) inner variables scaling; (b) outer variables scaling.

Inner and outer variable scaling are obtained using $u_\tau$ and $U_\infty$, respectively. The results are then compared with data by Choi & Moin (Reference Choi and Moin1990) and the model of Panton & Linebarger (Reference Panton and Linebarger1974). In figures 33(a,b), a good agreement is found for the ZPG distributions with Choi & Moin (Reference Choi and Moin1990), and in the low- to mid-wavenumber range ($k_x\delta$ below 20) with the model of Panton & Linebarger (Reference Panton and Linebarger1974) (see both dashed dotted lines). Indeed, according to Panton & Linebarger (Reference Panton and Linebarger1974), a convective velocity overlap law, similar to the friction mean velocity log-law, can be defined in the mid-wavenumber range, in which large and small structures are both important. This overlap law depending upon the friction Reynolds number is plotted here for ${Re}_{\tau }=178$ and ${Re}_{\tau }=185$ corresponding to sensors 7 and 9, respectively (see table 1). The inner variable scaling of figure 33(a) shows a good collapse of the spectra upon $k_x\approx 15/\delta$, with APG levels of the same order as the ZPG levels. In figure 33(b), for small-scale structures (or high $k_x$), the ratio $U_c/U_{\infty }$ is around $0.6$ for ZPG. It decreases in the APG region to $0.2$ and $0.1$ at sensors 21 and 24, respectively. For large-scale structures (or low $k_x$), the ratio $U_c/U_{\infty }$ is higher around $0.8$ for ZPG. It again decreases in the APG region to $0.7$ and $0.5$ at sensors 21 and 24, respectively. Due to the difficulty in extracting the maxima of the wavenumber-frequency spectra in a discrete frequency range, the distributions of convective velocities in figure 33 are quite scattered particularly at low wavenumbers (or at the lowest frequencies), which corresponds to the noisiest part of the spectra. Nevertheless, at lower $k_x$, the convection velocities can be seen to slightly decrease, probably because for those wavenumbers the contribution of the outer region of the boundary layer is lower and the middle region contribution picks up.

As already observed by Choi & Moin (Reference Choi and Moin1990), at high frequencies and wavenumbers, a discrepancy between $U_c(\omega )$ and $U_c(k_x)$ is observed: $U_c(\omega )\approx 0.8U_{\infty }$ and $U_c(k_x)\approx 0.6U_{\infty }$ for ZPG; and $U_c(\omega )\approx 0.5U_{\infty }$ and $U_c(k_x)\approx 0.2U_{\infty }$ for APG. This is probably due to the $\varPsi _{pp}(\boldsymbol {k};\omega )$ characterised by a higher aspect ratio at high frequencies and wavenumbers, as also shown in figure 9 in Choi & Moin (Reference Choi and Moin1990) (see § 4.3 and Wu et al. (Reference Wu, Moreau and Sandberg2019)). Overall, as discussed for $U_c(\omega )$, as $k_x$ increases, the decrease of $U_c(k_x)$ is more evident for APG than for ZPG, probably due to the smaller (Lee & Sung Reference Lee and Sung2008) and slower-moving eddies near the wall in an APG flow.

4.4.3. Convection velocity as a function of longitudinal separation

The convection velocity is also commonly defined from the cross-correlation $R_{pp}(\xi _x, \tau )$ as function of the streamwise separation as $U_c(\xi _x) =\xi _x/\tau _{max}(\xi _x)$ where the time delay $\tau _{max}$ is given by the maximum $[\partial R(\xi _x,\tau )/\partial \tau ]_{\tau =\tau _{max}(\xi _x)}=0$. In figure 34, the normalised convection velocities as function of longitudinal separation $\xi _x/\delta ^{*}$ are shown, together with the peaks of the longitudinal space–time correlation at the four locations on the aerofoil suction side.

Figure 34. (a) Normalised convection velocities as function of longitudinal separation and (b) peaks of the longitudinal space–time correlation at four locations on the aerofoil suction side: ZPG locations $x/c=-0.60$ (thin grey line) and $x/c=-0.47$ (thick grey line); APG locations $x/c=-0.14$ (thin black line), $x/c=-0.08$ (thick black line); $\bullet$, data by Choi & Moin (Reference Choi and Moin1990); $\circ$, ZPG data $x=0.5$, and $\square$, APG data $x=0.85$ for attached TBL and separated TBL (grey) at $x/\delta _{in}^{*}=120$ by Na & Moin (Reference Na and Moin1998); $\triangle$, ZPG (black), APGs (red) and APGw (magenta) data by Cohen & Gloerfelt (Reference Cohen and Gloerfelt2018); $\blacktriangle$, APG data by Burton (Reference Burton1973).

In figure 34(a), the ratio $U_c(\xi _x)/U_{e}$ is shown to increase almost monotonously for all data sets as the longitudinal separation increases. In fact, as a group of turbulent eddies travel downstream, the smaller eddies die out more rapidly and the larger eddies (with higher $U_c$ and a longer life time) dominate the wall-pressure correlation, as evidenced in figure 34(b). Indeed, as shown in figure 17, the time delay corresponding to the peaks of the longitudinal space–time correlation for a particular eddy is shorter in ZPG than in APG, so lower values of correlation and longer delay times are found in the APG flow. The relationship between APG and ZPG flows is similar to what it is found for $U_c(\omega )$ and $U_c(\xi _x)$. For a given value of $\xi _x/\delta ^{*}$ the convective velocity is lower for APG than for ZPG (Schloemer Reference Schloemer1967; Na & Moin Reference Na and Moin1998). In figure 34(a), the convection velocities at large separations $\xi _x$ are close to what is found at low $k_x$: approximately $0.7\,U_e$ at $\xi _x=11\delta ^{*}$ (or $0.85\,U_{\infty }$) for ZPG and approximately $0.5\,U_e$ at $\xi _x=5\delta ^{*}$ (or approximately $0.55\,U_{\infty }$) for APG. On the other side, the convection velocities at small $\xi _x$ separations are close to what is found at high $k_x$ (see figure 33b). A good agreement with the reference data is found for both APG and ZPG (Burton Reference Burton1973; Choi & Moin Reference Choi and Moin1990; Na & Moin Reference Na and Moin1998; Cohen & Gloerfelt Reference Cohen and Gloerfelt2018). This is also consistent with the APG region on the NACA0012 aerofoil as shown in figure 21 in Brooks & Hodgson (Reference Brooks and Hodgson1981). Figure 34 shows how an increase of Reynolds number is characterised by a faster decay in $\xi _x$ with the low-frequency wall-pressure fluctuations retained for extended streamwise distances. This is a characteristic of hairpin structures (Van Blitterswyk & Rocha Reference Van Blitterswyk and Rocha2017), or long coherent motions containing a large number of smaller events.

In summary, for all three definitions of the convection velocity in the present highly non-equilibrium turbulent flow, the APG is shown to significantly reduce the convection velocity (by a factor 2–3), consistently with previously measured ZPG and APG TBL. In terms of scaling, good collapse is found when all defined convection velocity are normalised by the friction velocity $u_\tau$.