2. Experimental methodology

2.1. Experimental facility

An experimental campaign was carried out at the Centre for International Cooperation in Long-Pipe Experiments (CICLoPE, see figure 1 a,b). The laboratory is realised inside a mountain to keep stable environmental conditions and to minimise background noise, while sound-absorbing material ensures minimal acoustic interference in the test section. The closed-loop facility comprises a 111.15 m-long circular pipe with a radius of $R = D/2 = 0.4505\,\rm m$ . The primary streamwise location for measurements (where the flow is fully developed) is at $x^\prime = 110.1\,\textrm { m} = 244.4R$ downstream of the pipe inlet. For the experiments reported, the pipe flow was operated at seven centreline velocities, with a maximum of $U_{{ CL}} = 44.60\,\textrm { m}\,{\rm s^-{^1}}$ (corresponding to $Re_\tau \equiv U_\tau R/\nu = 47\,015$ ). Test conditions are elaborated upon in § 2.3. For presenting results, a Cartesian coordinate system is adopted with its origin at the primary streamwise location for measurements (at the centre of sensor $\mathcal{M}_1$ , indicated in figure 1 c). Here, the $x$ -axis denotes the streamwise direction (positive in the downstream direction) and the $y$ -axis denotes the wall-normal direction ( $y = 0$ at the wall and positive towards the centreline of the pipe). A comprehensive description of all design details of the facility can be found in the literature (Talamelli et al. Reference Talamelli, Persiani, Fransson, Alfredsson, Johansson, Nagib, Rüedi, Sreenivasan and Monkewitz2009; Bellani & Talamelli Reference Bellani and Talamelli2016).

Figure 1.

(a) Photograph of the CICLoPE laboratory, with (b) the test section at the downstream end of the long-pipe facility. (c) Schematic of the microphone sensor placement ( $\mathcal{M}1$ to $\mathcal{M}4$ were mounted in the pipe wall and $\mathcal{M}5$ was mounted along the pipe centreline). (d) Illustration of the points in the area of interest where acquisitions with single-wire and x-wire probes were performed. (e) Schematic of the pinhole–sub-surface cavity, used to mount the microphones in the pipe wall.

2.3. Experimental conditions and measurement resolution

Seven experimental conditions were considered for measurements of the fluctuating wall-pressure and velocity in the CICLoPE long-pipe facility. Flow parameters of all test cases are reported in table 1. With the aid of a heat exchanger, the facility was operated at constant temperature and the angular velocities of the two co-axial fans were set to generate centreline velocities in the range $3.837\,\textrm { m}\,{\rm s^-{^1}} \leqslant U_{\textit{CL}} \leqslant 44.60\,\textrm { m}\,{\rm s^-{^1}}$ (measured with a Pitot-static probe). Corresponding values of the wall-shear stress, $\tau _w$ , were inferred from static pressure drop measurements (following Fiorini Reference Fiorini2017). Values for the air density were indirectly measured with the air flow temperature and barometric pressure, so that values for the friction velocity, $U_\tau$ , could be computed. For the experiments reported in this work, friction Reynolds numbers were in the range $4794 \lesssim Re_\tau \lesssim 47\,015$ .

Table 1.

Flow parameters corresponding to the seven test conditions in the CICLoPE long-pipe facility, alongside non-dimensional parameters of the instrumentation’s geometry and acquisition details.

Spatial and temporal resolutions need to be considered for both the fluctuating wall-pressure and velocity measurements. For the measurement of wall-pressure, the pinhole orifice diameter dictates the spatial resolution, while for the measurement of velocity, the hot-wire sensing length is determining the spatial resolution. The temporal resolution was limited by the acquisition frequency. All three parameters relevant for the measurement resolutions ( $d_p$ , $l_{hw}$ and $f_s$ ) are listed in table 1 after normalisation with the viscous scales.

For fully resolved wall-pressure measurements, the pinhole orifice diameter must be $d_p^+ \lt 20$ (Gravante et al. Reference Gravante, Naguib, Wark and Nagib1998). Hence, the pinhole diameter is not sufficiently small to claim fully resolved wall-pressure measurements for test cases 4 to 7 (the relatively large values of $d_p^+$ result in an attenuation of small-scale energy). However, this work does not revolve around conducting fully resolved measurements, but rather focuses on the correlation between velocity fluctuations in the logarithmic region and wall-pressure. As reviewed in § 1, the scales of interest for the correlation analyses reside at streamwise wavelengths of $\lambda _x/y \gtrsim 3$ (when considering $u$ fluctuations) and $\lambda _x/y \gtrsim 1$ (when considering $v$ fluctuations). Smaller streamwise scales in both the pressure and pressure-squared time series are not relevant, as they do not correlate with the ones in the turbulent velocity signals. Consequently, for all $Re_\tau$ test cases, a minimum streamwise wavelength that needs to be resolved for the coherence analyses is given by $\lambda _{x,{res}}/y_A = 1$ (recall that $y_A$ is the lowest wall-normal position being considered), resulting in a streamwise wavelength of $\lambda _{x,{res}} = y_A = 11\,\rm mm$ . The pinhole orifice diameter of ${d}_p = 0.3\,\rm mm$ is a factor of 36.6 smaller and, thus, sufficient for capturing the streamwise wavelengths of interest.

When considering the spatial resolution of the HWA measurements, a similar reasoning can be applied. Statistically, the velocity structures of relevance to the wall-pressure–velocity correlations adhere to a self-similar scaling in all three dimensions. Baidya et al. (Reference Baidya2019) showed that the aspect ratio of coherent velocity structures is 7 : 1, in terms of their characteristic streamwise-to-spanwise length scales. Hence, the smallest structures of relevance have spanwise wavelengths of $\lambda _{z,{res}} = \lambda _{x,{res}}/7 \approx 1.6\,\rm mm$ . For the HWA measurements with the x-wire probe, the spanwise separation between both sensing wires is ${\sim} 1.0\,\rm mm$ , which is sufficient to resolve the scales of interest. In the $y$ -direction, the sensing length of the x-wire probe is also adequate, given the strong wall-normal coherence of the velocity structures. For the single-wire probe, its spanwise sensing length of $l_{{ HW}} = 1.25\,\rm mm$ is more than sufficient given that the $u$ fluctuations of interest are three times larger than the $v$ fluctuations of interest.

Regarding the temporal measurement resolution, this is set by the data acquisition rate. For the highest $Re_\tau$ test case (number 7), the acquisition time step is largest in terms of viscous time scales and equals $\Delta T^+ = 1/f_s^+ \approx 3.3$ . Even though Hutchins et al. (Reference Hutchins, Nickels, Marusic and Chong2009) indicate a required time step $\Delta T^+$ of unity or less for fully resolved measurements, the current acquisition rate is more than sufficient given the interest in much lower frequencies (larger spatial scales) than those corresponding to the dissipative regime.

3. Wall-pressure statistics in the CICLoPE facility

Statistics of the wall-pressure fluctuations are presented to demonstrate the validity of our data for the correlation analyses presented in §§ 4 and 5.

Probability density functions (p.d.f.s) of the wall-pressure time series are shown in figures 2(a) and 2(b), with a linear and logarithmic scale on the ordinate axes, respectively. For both figures, the amplitude axes are scaled with the wall-pressure intensity (root-mean-square), denoted as $p^\prime _w$ . Superimposed are several p.d.f.s from the literature: a p.d.f. corresponding to an atmospheric boundary layer flow at $Re_\tau \approx 1 \times 10^6 \pm 2\times 10^5$ (Klewicki et al. Reference Klewicki, Priyadarshana and Metzger2008) and a band representing the spread of p.d.f.s corresponding to zero-pressure-gradient TBL flow at $1313 \lesssim Re_\tau \lesssim 3826$ (Tsuji et al. Reference Tsuji, Fransson, Alfredsson and Johansson2007). All p.d.f.s of the current datasets show negligible disparity between the test cases and are consistent with the distributions from the literature. Minor deviations appear in the tails of the p.d.f.s (figure 2 b), yet comparable with the degree of deviation in the work by Tsuji et al. (Reference Tsuji, Fransson, Alfredsson and Johansson2007) and without a noticeable monotonic trend with an increase in $Re_\tau$ .

Figure 2.

Probability density functions of the wall-pressure fluctuations in the CICLoPE facility for all $Re_\tau$ test cases considered (see table 1). Current data are compared with a p.d.f. obtained from atmospheric boundary layer data at $Re_\tau \approx 10^6$ (Klewicki et al. Reference Klewicki, Priyadarshana and Metzger2008), label K08), and a band representing the spread of p.d.f.s obtained from zero-pressure-gradient TBL data at $1{}313 \lesssim Re_\tau \lesssim 3{}826$ (Tsuji et al. Reference Tsuji, Fransson, Alfredsson and Johansson2007, label T07). A standard $\mathcal{N}(0,1)$ Gaussian distribution is added for reference. Probability density functions are plotted with (a) a linear scale and (b) a logarithmic scale on the ordinate axes. Data employed for plotting panels (a) and (b) are available in the supplementary material.

Figure 3.

(a,b) Pre-multiplied energy spectra of wall-pressure fluctuations, for all $Re_\tau$ test cases considered (an increase in colour intensity corresponds to an increase in $Re_\tau$ following test cases $1\rightarrow 7$ , listed in table 1), as a function of (a) the viscous scaled wavelength and (b) the outer-scaled wavelength. Note that the temporal spectra are plotted as spatial spectra by converting frequency into wavelength, using $\lambda _x \equiv U_c/f$ with $U_c^+ = 10$ . Vertical dashed lines in panel (b) indicate the minimum wavelength in our dataset for which wall-pressure–velocity correlations become appreciable when considering $u$ fluctuations ( $\lambda _x/y_A = 3$ ) and $v$ fluctuations ( $\lambda _x/y_A = 1$ ). (c,d) Gain of transfer kernel $H_r$ that characterises the pinhole–sub-surface cavity as described in § 2.2, including in (c) the gain of the raw kernel, $H_r^{\textit{exp}}$ (light grey line). Data employed for plotting panels (a) and (b) are available in the supplementary material.

Pre-multiplied energy spectra of wall-pressure fluctuations are shown in figures 3(a) and 3(b) for all values of $Re_\tau$ , with an inner-scaled and outer-scaled streamwise wavelength on the abscissa, respectively. Here, the streamwise wavelength is obtained by applying Taylor’s hypothesis: $\lambda _x \equiv U_c/f$ , where $f$ is the frequency and $U_c$ is the convection velocity taken as $U_c^+ = 10$ . Despite the convection velocity of the wall-pressure field being scale-dependent (e.g. Luhar, Sharma & McKeon Reference Luhar, Sharma and McKeon2014) and the temporal-to-spatial conversion of near-wall fluctuations in velocity/pressure not abiding by Taylor’s hypothesis (Dennis & Nickels Reference Dennis and Nickels2008; del Á lamo & Jiménez Reference del Álamo and Jiménez2009), the conversion is kept equal across all test cases. In essence, we compare temporal spectra (since $\lambda _x^+ = 10U^2_\tau /\nu /f = 10/f^+$ ). Still the temporal-to-spatial conversion was applied because §§ 4 and 5 consider all coherence spectra as a function of wavelength for ease of comparison to the only data available (those from spatial DNS of turbulent channel flow).

Before commenting on the wall-pressure spectra, note that figures 3(c) and 3(d) present the gain of the transfer kernel that characterises the pinhole–sub-surface cavity (described in § 2.2). Because the transfer kernel is a function of frequency and the frequency-to-wavelength conversion includes the friction velocity of each test case, seven identical kernels are shown (but shifted along $\lambda _x$ ). For reference, the raw experimental kernel, $H_r^{{ exp}}$ , is shown in figure 3(c) for the highest $Re_\tau$ test case with a thick grey line, while the other kernels correspond to the fitted kernel of the second-order model, $H_r$ . Noticeably, resonance occurs at scales where the wall-pressure spectra are energetic. It is thus necessary to correct the spectra for the amplification/attenuation effect. Current wall-pressure spectra were corrected before plotting, by dividing the spectra with the frequency-dependent model kernel ( $\phi _{pp}(f) = \phi _{pp,f}(f)/\vert H_r(f)\vert ^2$ , with $\phi _{pp,f}$ being the spectrum after removing facility noise from the raw measurements of wall-pressure following Appendix A). The resonance-correction works theoretically, but practically the kernel (which was found with the aid of a flow-off experiment) changes when wall-bounded turbulence grazes the pinhole orifice (see Dacome et al. Reference Dacome, Mörsch, Kotsonis and Baars2024b ), making the correction imperfect. In practice, this results in an erroneous ‘wiggle’ in various spectra and is most noticeable in the spectrum of test case 7.

Close inspection of the wall-pressure spectra reveals expected Reynolds-number trends. At first, the location of the inner-spectral peak at $\lambda _{x,p}^+ \approx 250$ (figure 3 a) agrees well with previous findings (Farabee & Casarella Reference Farabee and Casarella1991; Tsuji et al. Reference Tsuji, Fransson, Alfredsson and Johansson2007; Klewicki et al. Reference Klewicki, Priyadarshana and Metzger2008; Panton et al. Reference Panton, Lee and Moser2017). A slight increase in the inner-spectral peak magnitude, with an increase in $Re_\tau$ , is also noticeable for test cases 4 to 7 (expected per the trends in Tsuji et al. Reference Tsuji, Fransson, Alfredsson and Johansson2007; Panton et al. Reference Panton, Lee and Moser2017; Yu et al. Reference Yu, Ceci and Pirozzoli2022). The large-scale energy content also progressively increases with $Re_\tau$ and exhibits a collapse in outer-scaling, for the range $\lambda _x/R \gtrsim 0.2$ (figure 3 b). This trend is in line with the findings reported in DNS studies at lower $Re_\tau$ (Panton et al. Reference Panton, Lee and Moser2017; Yu et al. Reference Yu, Ceci and Pirozzoli2022). It also conforms to the work by Deshpande et al. (Reference Deshpande, Vinuesa, Klewicki and Marusic2025), who reason that the intermediate and large scales of the wall-pressure spectra grow with $Re_\tau$ due to the contributions of both the active and inactive motions in the grazing flow. Spectra corresponding to test cases 1 and 2 are outliers in that their broadband peak-magnitudes are relatively high. We postulate that this is due to an incomplete removal of facility noise, as any remaining signature of facility noise is more pronounced in the spectra of lower $Re_\tau$ test cases. That is, the degree of facility noise was quantified with a signal-to-noise ratio (SNR), defined as the intensity ratio of turbulence-induced wall-pressure fluctuations, relative to those induced by facility noise: $ { SNR} = p^\prime _w/(p^\prime _{w,r} - p^\prime _w)$ . Here, $p^\prime _{w,r}$ is the pressure intensity (root-mean-square) of the raw, measured wall-pressure. SNRs in our dataset increase monotonically with $Re_\tau$ , in the interval $0.08 \leqslant { SNR} \leqslant 0.25$ . Additive facility noise is thus more noticeable in the spectra at lower $Re_\tau$ . For the remainder of the paper, it is important to recall from § 2.3 that for the correlation analysis, the scales of interest reside at streamwise wavelengths beyond $\lambda _x/y \approx 3$ (when considering $u$ fluctuations) and $\lambda _x/y \approx 1$ (when considering $v$ fluctuations). Both of these limits are indicated in figure 3(b); within the scale-range of interest, the spectra are not affected by the kernel-correction and only the two lowest test cases seem affected by additive (acoustics-driven) noise.

Figure 4.

Wall-pressure intensity inferred from integrating the wall-pressure spectra. Current results are compared with several datasets available from the literature. Data are taken from the DNS studies of Panton et al. (Reference Panton, Lee and Moser2017) (P17-DNS, $\bullet$ , ZPG-TBL), Choi & Moin (Reference Choi and Moin1990) (CM90-DNS, $\blacktriangledown$ , TCF) and Yu et al. (Reference Yu, Ceci and Pirozzoli2022) (YU-DNS, $\blacksquare$ , pipe flow). Furthermore, data are collected from experimental studies of ZPG-TBL flows: Blake (Reference Blake1970) (B70, $\triangle$ ), Bull & Thomas (Reference Bull and Thomas1976) (BT76, $\triangleleft$ ), Farabee & Casarella (Reference Farabee and Casarella1991) (FC91, $\triangleright$ ), Horne (Reference Horne1989) (H89, $\square$ ), Klewicki et al. (Reference Klewicki, Priyadarshana and Metzger2008) (K08, ), McGrath & Simpson (Reference McGrath and Simpson1987) (MS87, ), Schewe (Reference Schewe1983) (S83, ) and Tsuji et al. (Reference Tsuji, Fransson, Alfredsson and Johansson2007) (T07, $\circ$ ), and of experimental studies of pipe flows: Lauchle & Daniels (Reference Lauchle and Daniels1987) (LD87, ) and Morrison (Reference Morrison2007) (M07, ). Solid and dashed lines are the formulations presented by Klewicki et al. (Reference Klewicki, Priyadarshana and Metzger2008), in which the pressure variance increases logarithmically with increasing $Re_\tau$ . Data employed for plotting the wall-pressure intensity corresponding to the present work are available in the supplementary material.

As a final wall-pressure statistic, we consider the wall-pressure intensities, resulting from the integration of the energy spectra. Here, the root-mean-square intensity is considered and inner-normalised following $p^{\prime +}_w = p^{\prime }_w/\tau _w$ . Wall-pressure intensities are plotted in figure 4 and compared with a variety of datasets from the literature. Data from channel flow DNS are added (Panton et al. Reference Panton, Lee and Moser2017), together with the various datasets assembled by Klewicki et al. (Reference Klewicki, Priyadarshana and Metzger2008) (and named in the caption) that include both numerical and experimental studies, comprising zero-pressure-gradient turbulent boundary layer (ZPG-TBL), turbulent channel (TCF) and pipe flows. Our current data confirm the trend of increasing pressure intensity with $Re_\tau$ and closely follow the empirical relation of Klewicki et al. (Reference Klewicki, Priyadarshana and Metzger2008). Only the data point of test case 1 (at $Re_\tau \approx 4{}794$ ) is an outlier, which is ascribed to the imperfect facility noise-filtering causing an overestimation of the wall-pressure intensity.

4. Coherence between streamwise velocity and wall-pressure fluctuations

To analyse the scale-dependent coupling between the fluctuations in streamwise velocity ( $u$ ) and wall-pressure ( $p_w$ ), the linear coherence spectrum (LCS) is employed. The LCS describes the stochastic coupling, on a per-scale basis, as the degree of phase-consistency. The LCS is defined as the magnitude-squared of the cross-spectrum between $u$ and $p_w$ , normalised with the two auto-spectra of $u$ and $p_w$ :

(4.1) \begin{equation} \gamma _{up_w}^2\left (y,\lambda _x\right) \equiv \frac {\vert \langle \widetilde {U}\left (y,\lambda _x\right) \widetilde {P}^*_w\left (\lambda _x\right) \rangle \vert ^2}{\langle \vert \widetilde {U}\left (y,\lambda _x\right) \vert ^2 \rangle \langle \vert \widetilde {P}_w\left (\lambda _x\right)\vert ^2 \rangle }, \end{equation}

where the angled brackets $\langle \ldots \rangle$ indicate ensemble averaging, the $*$ denotes the complex conjugate and capitalised variables with a tilde indicate the Fourier transformed quantity, e.g. $\widetilde {P}_w(f) = \mathcal{F} [p_w(t)]$ . Because in the remainder of the manuscript we present scale-dependent data as a function of streamwise wavelength, the argument in (4.1) is taken as $\lambda _x$ and is, as for the energy spectra in § 3, obtained by applying Taylor’s hypothesis: $\lambda _x \equiv U_c/f$ , with $U_c^+ = 10$ .

Figure 5.

(a) Coherence spectra for the fluctuations in streamwise velocity and wall-pressure, and (b) the streamwise velocity and wall-pressure-squared. Two sets of coherence spectra are shown, corresponding to velocity fluctuations measured at point $A$ (blue colour scale) and point $F$ (red colour scale); an increase in colour intensity corresponds to an increase in $Re_\tau$ following test cases $1\rightarrow 7$ , listed in table 1. Reference data are shown with a light grey shaded area, associated with the spread of coherence spectra from spatial DNS data at $Re_\tau = 5{}200$ (Baars et al. Reference Baars, Dacome and Lee2024). (c) Coherence spectra for the fluctuations in streamwise velocity and wall-pressure, and (d) the streamwise velocity and wall-pressure-squared, for test case 3 ( $Re_\tau \approx 14\,004$ ), and for velocity fluctuations measured at points E, A–D spanning a range of streamwise locations, $-0.07 \leqslant x/R \leqslant 0.67$ . Note that all current coherence spectra are generated from temporal data and plotted as spatial spectra by converting frequency into wavelength using $\lambda _x \equiv U_c/f$ with $U_c^+ = 10$ . Data employed for plotting panels (a) and (b) are available in the supplementary material.

Figures 5(a) and 5(b) present the LCS for $u$ and $p_w$ , for two positions of the velocity measurement (points $A$ and $F$ in figure 1 d) and for all values of $Re_\tau$ . In presenting the scale-dependent spectra, we resort to scaling $\lambda _x$ with the distance-from-the-wall, so that the abscissae are in terms of $\lambda _x/y$ with $y = \{y_{A},y_{F}\}$ for these graphs. With negligible coherence reported at small wavelengths, a steady rise in the LCS can be observed in figure 5(a) for increasing $\lambda _x/y$ until a local maximum is reached at $(\lambda _x/y,\gamma _{up_w}^2) \approx (14,0.1)$ . Given that the current data span nearly a decade in $Re_\tau$ , we can conclude that the region of coherence centred at $\lambda _x/y \approx 14$ is Reynolds-number invariant. Only the two LCS corresponding to test cases 1 and 2 have a slightly lower value near $\lambda _x/y = 14$ . This is ascribed to the incomplete removal of facility noise in the wall-pressure spectra of these two test cases (recall the discussion in § 3), and the fact that additive noise in the spectra causes an attenuation of the LCS. Only a slight discrepancy in the $\lambda _x/y$ locations of the local maxima in coherence appears between the sets of coherence spectra corresponding to point A (blue colour scale) and point B (red colour scale). The differences in the $\lambda _x/y$ locations of the local maxima are marginal in comparison to the difference in wall-normal position ( $y_F \approx 6y_A$ ). In fact, there is no evidence that the locations of the maxima depend on the wall-normal location being considered, as the coherence spectra from DNS data – corresponding to various wall-normal locations being considered – show excellent collapse when $\lambda _x$ is scaled with the distance-from-the-wall (Baars et al. Reference Baars, Dacome and Lee2024).

A second region of significant coherence appears at large wavelengths. This region manifests itself by a Reynolds-number-invariant rise in coherence (from approximately the wavelengths indicated with the vertical dotted lines in figure 5 a), up to wavelengths where the velocity fluctuations continue to be energetically relevant. To illustrate this, an amplitude threshold of the pre-multiplied streamwise energy spectra is taken as $k^+_x\phi ^+_{uu} = 0.2$ at the large-scale end. Energy levels only drop below this threshold for outer-scaled wavelengths of $\lambda _{x,{lim}}/R \gtrsim 35$ . This limit is included in figure 5(a) for reference and corresponds to $\lambda _{x,{lim}}/y \approx \{250,1{}500\}$ for $y = \{y_A,y_F\}$ , respectively.

This large-scale region of strong coherence between $u$ fluctuations in the logarithmic region and the wall-pressure field is presumably related to global velocity modes (Bullock, Cooper & Abernathy Reference Bullock, Cooper and Abernathy1978; del Á lamo &Jiménez Reference del Álamo and Jiménez2003). These global modes are ‘inactive’ in the view of Townsend’s attached-eddy hypothesis (Townsend Reference Townsend1976) (thus, large-scale eddies that do not contribute to the Reynolds shear stress $uv$ ). Inactive motions are coupled to the very large scales in the pressure spectrum ( $TU_\infty /\delta \gt 7$ , with $T$ being their characteristic period and $\delta$ the boundary layer thickness) (as shown explicitly by Deshpande et al. Reference Deshpande, Vinuesa, Klewicki and Marusic2025), while the active motions contribute directly to the intermediate scales ( $0.8\lt TU_\infty /\delta \lt 7$ ). The limit of $TU_\infty /R=7$ is identified in figure 5(a), for the measurements corresponding to point $y_A$ and point $y_F$ , by means of the blue and red vertical dotted lines at the corresponding $\lambda _x/y$ values, respectively.

Before further discussing the trends of the coherence spectra, we proceed with inspecting the coherence involving the quadratic term of the wall-pressure. The inclusion of this term was deemed important for stochastically estimating off-the-wall velocities from wall-pressure data. The quadratic term of the wall-pressure is taken as $p_w^2 = [p_w^2]_{r} - \overline { [p_w^2]_{r}}$ , with $ [p_w^2]_{r}$ denoting the time series of the wall-pressure-squared prior to the subtraction of its mean. Different to the behaviour displayed by the linear term of wall-pressure, the LCS for $u$ and $p_w^2$ rises starting from $\lambda _x/y \approx 7$ (see figure 5 b). Again, a Reynolds-number-invariant trend appears in the rise of coherence around scales of $\lambda _x/y = 14$ and beyond, with once more the LCS of test cases 1 and 2 comprising a lower magnitude due to the incomplete removal of facility noise from the wall-pressure spectra. To further conclude the Reynolds-number-invariant trends observed in figure 5(a,b), the current experimental coherence spectra generated from temporal data are compared with the coherence spectra presented by Baars et al. (Reference Baars, Dacome and Lee2024), generated from spatial DNS data of turbulent channel flow. These reference data are shown with the light grey shaded area, indicating the spread of coherence spectra at $Re_\tau = 5{}200$ when considering a range of wall-normal positions across the logarithmic region ( $80 \lesssim y^+ \lesssim 0.15Re_\tau$ , see figure 6 of Baars et al. Reference Baars, Dacome and Lee2024). Moreover, Baars et al. (Reference Baars, Dacome and Lee2024) also revealed a Reynolds-number-invariant trend for these DNS data, spanning $550\lesssim Re_\tau \lesssim \approx 5200$ . It must be noted that even though these DNS data are associated with turbulent channel flow, it was shown that coherence spectra from a relatively low-Reynolds-number TBL flow ( $Re_\tau \approx 2280$ ) were also in agreement with these channel flow data. Therefore, with the current LCS for pipe flow collapsing for the full range of $Re_\tau$ , for both wall-normal positions, $y = \{y_{A},y_{F}\}$ (while agreeing with the reference data), it can be concluded that the coherence is statistically similar across several canonical flow geometries.

Universal trends in the coherence spectra are reflective of how coherent velocity fluctuations are interlinked to the wall-pressure. The scales around which $\gamma ^2_{up_w}$ and $\gamma ^2_{up^2_w}$ become non-zero ( $\lambda _x/y \approx 3$ and $\lambda _x/y \approx 7$ , respectively), as well as the logarithmic growth of coherence (most noticeable in figure 5(b), where $\gamma ^2_{up^2_w} \propto \ln (\lambda _x/y)$ ) follow a pattern presented in the work by Baars et al. (Reference Baars, Hutchins and Marusic2017). They considered the coherence between the near-wall velocity fluctuations and those in the logarithmic region. The logarithmic growth of coherence, which occurs over the inertial range of wavelengths, was interpreted as the range of scales that contains turbulence energy that is statistically self-similar (following a hierarchical structure of wall-attached eddies).

Figure 6.

(a) Normalised wall-pressure time series of microphone $\mathcal{M}1$ , for test case 3 ( $Re_\tau \approx 14\,004$ ), its Hilbert transform and the corresponding de-meaned wall-pressure-squared. (b) Coherence spectra for the fluctuations in streamwise velocity and wall-pressure-squared (dashed lines, identical to the coherence spectra in figure 5 b), compared with the coherence spectra for fluctuations in the streamwise velocity at point $y_A$ and the Hilbert transform of the wall-pressure (solid lines).

An increase in large-scale coherence for the wall-pressure-squared term suggests that large-scale $u$ fluctuations modify (modulate) the wall-pressure field following nonlinear dynamics. To analyse this phenomenon, a Hilbert transform is used to retrieve an ‘envelope’ of the wall-pressure time series. Figure 6(a) presents the normalised wall-pressure time series ( $\widetilde {p}_w = p_w/p^{\prime }_w$ ) at $Re_\tau \approx 14\,004$ for microphone $\mathcal{M}1$ , over a short time interval, together with the magnitude of its Hilbert transform, $\vert H(\widetilde {p}_w) \vert$ , and the de-meaned wall-pressure-squared time series. By visual inspection, these last two time series have similar large-scale energy content. Figure 6(b) quantifies this further by overlaying the LCS for $u$ and $\vert H(\widetilde {p}_w)\vert$ , and the LCS for $u$ and $p_w^2$ . A remarkable collapse is observed for the two sets of LCS spectra, for all $Re_\tau$ cases considered. Hence, the large-scale variations in the wall-pressure intensity are directly linked to the passage of streamwise velocity fluctuations modulating the near-wall intensity (Tsuji et al. Reference Tsuji, Marusic and Johansson2015).

Gaining knowledge on how the coherence decays as a function of the streamwise separation between the velocity measurement and the wall-pressure sensor is highly relevant for real-time flow control (e.g. when sensors and actuators are separated to allow for control actions while the flow convects downstream). For our current data, the $\gamma ^2_{up_w}$ and $\gamma ^2_{up^2_w}$ coherence spectra are considered as a function of the streamwise distance of the velocity measurement (relative to the wall-pressure sensor at $x = 0$ ) for test case 3 corresponding to $Re_\tau \approx 14\,004$ . Coherence spectra are shown in figures 5(c) and 5(d), for $\gamma ^2_{up_w}$ and $\gamma ^2_{up^2_w}$ , respectively. When increasing the streamwise distance, $\gamma ^2_{up_w}$ decays with the coherence decreasing faster at smaller scales, as is expected. Similar conclusions were drawn for all other $Re_\tau$ test cases. When inspecting the decay in $\gamma _{up_w^2}^2$ (figure 5 d), it becomes clear that the coherence with the quadratic wall-pressure term remains considerably larger than with the linear term. This means that the mechanism of large-scale modulation of the smaller-scale wall-pressure fluctuations (by the large-scale $u$ fluctuations) is dominant over the direct (linear) imprint of $u$ fluctuations on the wall-pressure.

5. Coherence between wall-normal velocity and wall-pressure fluctuations

To further characterise the dynamics between fluctuations in velocity and wall-pressure, the foregoing presents a similar analysis as described in § 4, but instead of considering the $u$ fluctuations, we concentrate on the $v$ fluctuations. Coherence spectra for $v$ and $p_w$ (figure 7 a) exhibit local maxima around $\lambda _x/y \approx 10$ (instead of $\lambda _x/y \approx 8$ , as indicated by Baars et al. Reference Baars, Dacome and Lee2024) and $\gamma _{vp_w}^2\approx 0.15$ , a magnitude which is roughly 50 % higher than local maxima in the LCS for $u$ and $p_w$ . Squaring the wall-pressure fluctuations brings higher levels of $\gamma _{vp_w^2}^2$ (see figure 7 b), starting to rise at $\lambda _x/y \approx 3$ . A similar Reynolds-number independence is observed as was seen in the coherence analyses with the $u$ fluctuations.

Figure 7.

(a) Coherence spectra for the fluctuations in wall-normal velocity and wall-pressure, and (b) the wall-normal velocity and wall-pressure-squared. Two sets of coherence spectra are shown, corresponding to velocity fluctuations measured at point $A$ ; an increase in colour intensity corresponds to an increase in $Re_\tau$ following test cases $1\rightarrow 7$ , listed in table 1. The light grey shaded area is associated with the spread of coherence spectra from DNS data, as reported by Baars et al. (Reference Baars, Dacome and Lee2024). Note that all current coherence spectra are generated from temporal data and plotted as spatial spectra by converting frequency into wavelength using $\lambda _x \equiv U_c/f$ with $U_c^+ = 10$ . Data employed for plotting panels (a) and (b) are available in the supplementary material.

Higher coherence between wall-normal velocity and wall-pressure fluctuations complies with earlier findings (e.g. Gibeau & Ghaemi Reference Gibeau and Ghaemi2021). In fact, given the non-permeability boundary condition, fluid motions directed towards the wall (with a negative $v$ -component) will stagnate and thus give rise to a positive fluctuation in wall-pressure. Conversely, $v \gt 0$ fluctuations will tend to lower the unsteady wall-pressure. In general, the higher correlation between $v$ and $p_w$ can be ascribed to the Orr mechanism (see Jiménez Reference Jiménez2013; Luhar et al. Reference Luhar, Sharma and McKeon2014, among others). Therefore, there exists a rather strong coupling between $v$ and $p_w$ . Note, however, that our current work is motivated by using wall-pressure information to, eventually, predict the off-the-wall velocity fluctuations. Using pressure information to predict wall-normal velocity fluctuations for real-time control is not as effective as when considering streamwise velocity fluctuations, because the streamwise ones provide a stronger contribution to wall-shear stress generating mechanisms (e.g. Deck et al. Reference Deck, Renhard, Laraufie and Weiss2014, among others). Furthermore, $v$ fluctuations have a considerably shorter characteristic wavelength than $u$ fluctuations ( $\lambda _x^+ \approx 250$ for the former and $\lambda _x^+ \approx 1000$ for the latter). This not only requires faster processing for real-time operations, but would also constrain the sensor-actuator spacing due to the faster de-correlation of the $v$ fluctuations in the streamwise direction.

6. Stochastic estimation of streamwise velocity fluctuations

For real-time control purposes, wall-pressure sensing can be employed to generate an estimate of the fluctuating velocities in the logarithmic region of a wall-bounded turbulent flow. To this end, the foregoing will examine the accuracy of the prediction of $u$ fluctuations, performed with Linear and Quadratic Stochastic Estimation methods (LSE and QSE, respectively). These methods solely employ the time series of wall-pressure and wall-pressure-squared, as the input quantities.

Estimates of the $u$ fluctuations in the logarithmic region at position $y_e$ can be formed through a convolution of time-domain kernels. Estimates of unconditional time series of off-the-wall velocity fluctuations, with the LSE and QSE procedures, follow from the formulations:

(6.1) \begin{eqnarray} \widehat {u}_{\textit{LSE}}\left (y_e,t\right) &=& \left (h_l \circledast p_w \right)\left (t\right), \\[-27pt] \nonumber \end{eqnarray}

(6.2) \begin{eqnarray} \widehat {u}_{\textit{QSE}}\left (y_e,t\right) &=& \left (h_l \circledast p_w \right)\left (t\right) + \left (h_q \circledast p^2_w \right)\left (t\right), \end{eqnarray}

where the (stochastic) temporal kernels of the linear term ( $h_l$ ) and quadratic term ( $h_q$ ) are the inverse Fourier transforms of the complex, frequency-domain kernels, e.g. $h_l (y_e,t) = \mathcal{F}^{-1} [H_L (y_e,f)]$ and similar for $h_q$ . The linear kernel equals the cross-spectrum between $u$ and $p_w$ , divided by the auto-spectra of $p_w$ (the input quantity during the estimation method),

(6.3) \begin{equation} H_L\left (y_e,f\right) = \frac {\langle \widetilde {U}\left (y_e,f\right) \widetilde {P}^*_w\left (f\right) \rangle }{\langle \vert \widetilde {P}_w\left (f\right) \vert ^2 \rangle }, \end{equation}

whereas the quadratic kernel includes the wall-pressure-squared term, $p_w^2$ , instead of the linear one,

(6.4) \begin{equation} H_Q\left (y_e,f\right) = \frac {\langle \widetilde {U}\left (y_e,f\right) \widetilde {P}^*_{w,sq}\left (f\right) \rangle }{\langle \vert \widetilde {P}_{w,sq}\left (f\right) \vert ^2 \rangle }, \end{equation}

with $\widetilde {P}_{w,sq} (f) = \mathcal{F} [p^2_w(t)]$ . Given our current dataset, two estimation locations are considered ( $y_e = y_A$ and $y_e = y_F$ ). Further details of the stochastic estimation procedures can be found elsewhere (Naguib et al. Reference Naguib, Wark and Juckenhöfel2001; Baars et al. Reference Baars, Dacome and Lee2024).

To evaluate the accuracy of the estimation with respect to the reference time series, $u(y_0,t)$ , the Pearson correlation coefficient is employed. It is defined as the ratio of the covariance of two input signals to the product of the standard deviation of the two. Figure 8(b) presents values of $\rho [u_W(y,t),\widehat {u}_{{ QSE}}(y,t)]= \mathrm{cov} [u_W(y,t),\widehat {u}_{{ QSE}}(y,t)]/(u_W^{\prime} \widehat{u}^{\kern1pt \prime}_{{ QSE}})$ : the correlation coefficient between the reference time series $u_W(y,t)$ at points $A$ and $F$ (see figure 1 b) to the QSE-based time series, $\widehat {u}_{{ QSE}}(y,t)$ . Here, the time series $u_W(y,t)$ is not equal to $u(y,t)$ , because $u_W$ only retains wall-attached eddies. Effectively, $u_W$ is a large-scale pass-filtered signal of $u$ , with its Reynolds number-invariant kernel characterised by a definitive cutoff at $\lambda _x/y = 14$ (Baars et al. Reference Baars, Hutchins and Marusic2017).

Figure 8.

(a) Correlation coefficient computed with the QSE-based streamwise velocity fluctuations in the logarithmic region, $\widehat {u}_{{ QSE}}$ , and the reference time series, $u_W$ . (b) Correlation coefficient computed with the LSE-based streamwise velocity fluctuations in the logarithmic region, $\widehat {u}_{{ LSE}}$ , and the reference time series, $u_W$ . Reference data are taken from Baars et al. (Reference Baars, Dacome and Lee2024) at $Re_\tau \approx 2300$ .

With the exception of the data at the two lowest $Re_\tau$ test cases 1 and 2 – whose time series are still affected by imperfect noise filtering (see § 2.4) – we observe collapse for all $Re_\tau$ test cases considered. A slight attenuation of $\rho$ is observed with respect to the reference data, which is attributed to a systematic error (Saccenti et al. Reference Saccenti, Hendriks and Smilde2020) caused by additive facility noise. Furthermore, the data are in good agreement with the reference data from Baars et al. (Reference Baars, Dacome and Lee2024) at $Re_\tau \approx 2300$ . This result solidifies the conclusion of Reynolds-number-independent estimation accuracy of wall-attached velocity fluctuations, based on wall-pressure sensing alone. As expected, a lower correlation coefficient appears in figure 8(b), where the estimation is performed with only the linear term of wall-pressure (thus, using the LSE method). The performance of the statistical estimator now decays by approximately 20 % with respect to QSE-based estimation, across the given range of wall-normal distances.