2.1. Pressure measurement system

Three $1/4~\text{in.}$ microphones (combination of B&K microphone 4938 and B&K pre-amplifier 2670) were installed in the wind tunnel. Microphone No. 1 with the pressure probe was attached to the traversing system, microphone No. 2 was mounted on the wall and microphone No. 3 with a nose cone (B&K UA-0385) was fixed in the free stream. These microphones were connected to the signal conditioner (B&K Nexus Range of Conditioning Amplifiers Type 2690) with a built-in filter. The cutoff frequencies of the high-pass and low-pass filters were set to 0.1 Hz and 10 kHz respectively. The signals were recorded by a 16 bit A/D converter board (Measurement Computing PCI-DAS6034) installed on a PC. The sampling rate was set at 40 kHz, and the Q-switch signals of the laser were simultaneously recorded for synchronization.

The design of the pressure probe and the wall pressure tap are shown in figures 4 and 5, and their dimensions are summarized in table 2. The probe dimensions in wall units at three Reynolds numbers are indicated in table 3. The pressure probe consists of the tip, pipe and connecting part to the microphone. The outer diameter of the stainless-steel pipe is 1.0 mm and the thickness is 0.05 mm. Two 0.4 mm diameter holes are opened with an angle of $180^{\circ }$ at 19.5 mm from the tip. The tip part has a conical shape of 10.0 mm length to minimize the flow disturbance. The pipe is glued to the connecting part and the distance from the two pressure holes to the microphone cavity is 30.2 mm. Consequently, the present pressure probe has a measurement volume of $0.4~\text{mm}\times 0.4~\text{mm}\times 1.0~\text{mm}$ in the $x{-}y{-}z$ directions. The design of the probe is the same as in Naka et al. (Reference Naka, Omori, Obi and Masuda2006) except for the number of pressure holes. The pressure probe was fixed to a support with a streamlined swept back shape. The centre of the two holes was positioned at the origin in the $x{-}z$ plane, and the probe axis was aligned to the wind-tunnel streamwise axis. The roll angle of the pressure probe in the wind tunnel was adjusted so that the centres of the two pressure holes were positioned at the same wall distance.

Figure 4. Schematic of the pressure probe (dimensions in mm).

Figure 5. Schematic of the wall pressure tap, section in the $y{-}z$ plane (dimensions in mm).

The wall pressure tap has a 0.5 mm diameter hole drilled with an inclination angle of $22^{\circ }$ in the spanwise direction, as shown in figure 5. This specific design is intended to place the wall pressure tap as close as possible to the stereo PIV measurement plane without optical/acoustic interference. The microphone is fixed directly inside the pressure probe or the wall, as close as possible to the tap. Thanks to the large scale of the boundary layer, a good spatial resolution is easily achieved. The diameter of the pressure sensing hole, in wall units, is $3.7^{+}$ , $6.3^{+}$ and $11.2^{+}$ for $U_{e}=3~\text{m}~\text{s}^{-1},5~\text{m}~\text{s}^{-1},10~\text{m}~\text{s}^{-1}$ respectively. It should be noted that these spatial resolutions are substantially better than the criteria for capturing small-scale pressure fluctuations as given by Schewe (Reference Schewe1983) or Gravante et al. (Reference Gravante, Naguib, Wark and Nagib1998).

The overall frequency response of the pressure measurement system is determined as a result of the acoustic and electrical transfer functions of individual components. This response is not flat. The low-frequency limit comes from the characteristics of the microphone and the pre-amplifier. In the present case, the microphone has a limiting frequency of approximately 1.5 Hz which gives $-$ 3 dB attenuation compared with the reference frequency. In addition, the pre-amplifier also produces attenuation and a phase shift in the frequency range below 10 Hz. The transfer functions of the microphone and amplifier provided by the manufacturer are used for compensating the pressure signals. On the other hand, the high-frequency response is limited by the acoustic resonance inside the pressure probe cavity which is typically several kHz. This resonance frequency can be optimized by choosing the dimensions of the probe: a shorter and thicker tube gives a higher resonance frequency and less damping, but this choice would cause a flow disturbance problem. The acoustic transfer function of each component was characterized based on a dynamic calibration in a similar manner to that described in Tsuji et al. (Reference Tsuji, Fransson, Alfredsson and Johansson2007) and Naka (Reference Naka2009). The white noise sound was recorded by the microphones with and without the pressure probe. The resonance frequencies in the measured transfer function are 1.7 and 2.2 kHz for the probe and wall tap. They are in good agreement with the values given analytically of 2.2 and 1.9 kHz. The amplitude response is flat within $\pm 3~\text{dB}$ up to 950 Hz for the pressure probe and within $\pm 1~\text{dB}$ up to 950 Hz for the wall pressure tap. The acoustic and electrical transfer functions are taken into account to recover reliable time series of the pressure fluctuation.

The signal-to-noise ratio of the pressure signal was enhanced by the use of auxiliary microphone No. 3 which was settled in the free stream. A Wiener noise canceller, described in Heyes (Reference Heyes1996), was implemented to reduce background acoustic/infra-sound noises which are commonly included in the signal from the probe and the free-stream microphones. In this case, data from two microphones are used for the noise reduction. One microphone is placed in the turbulent boundary layer which is sensitive to both turbulent and undesired non-turbulent pressure fluctuations. The other microphone is fixed in the free stream which is assumed free of turbulent pressure fluctuation. Our aim is to remove the noise, i.e. non-turbulent pressure fluctuations, of the first microphone. If the non-turbulent pressure fluctuations of two different microphones are identical, simple subtraction should recover the noise-free turbulent pressure fluctuation. In fact, this is not true for most cases. The signal of the free-stream microphone is used to estimate the non-turbulent pressure fluctuation commonly existing in both microphones. A time domain non-causal Wiener noise canceller with a filter order of $n=20\,000$ was used for noise reduction. The noise-reduced pressure fluctuations are used for the statistical evaluation shown in the following sections.

Figure 6(a,b) shows the instantaneous signals of pressure at $U_{e}=5~\text{m}~\text{s}^{-1}$ captured by the probes positioned at $y=3.8~\text{mm}$ and at the wall. Here, in order to demonstrate the effect of the noise reduction procedure, the noise-reduced signal and the raw signal with the free-stream pressure subtracted are compared. The perturbation around 6 Hz, due to the rotation period of the wind-tunnel blower, which is clearly observed in the signal of microphone No. 3 (not shown), cannot be seen in either signal of figure 6. This fan noise could be removed by simple subtraction, as acoustic contributions are supposed to contribute linearly. The power spectra of these signals are presented in figure 7. The profile of the subtracted signal still exhibits a peak at 6 Hz and is bumpy from 70 Hz to 2 kHz. The noise canceller works significantly better in the low-frequency range (below 150 Hz) and reduces the noise level in the high-frequency range ( ${\sim}1~\text{kHz}$ ) as well. The noise-reduced signals show a smooth profile up to approximately 200 Hz, and overall the power decreases monotonically towards high frequency. In practice, it was observed that it was much more efficient to use the Wiener noise canceller directly. The black curves in figures 6 and 7 show the representative performance of this filtering procedure.

Figure 6. Time series of pressure for 0.5 s: grey, raw signal with simple subtraction; black, noise-reduced signal; (a) pressure measured by the probe; (b) wall pressure.

Figure 7. Power spectral density (PSD) of pressure fluctuations: grey solid line, subtracted signal of probe pressure; grey dashed line, subtracted signal of wall pressure; black solid line, noise-reduced signal of probe pressure; black dashed line, noise-reduced signal of wall pressure.

2.2. Stereo PIV set-up

As depicted in figure 3, two stereo PIV planes were arranged in the $y{-}z$ plane and placed adjacent to each other in the wall-normal direction to cover the whole boundary layer thickness with a good spatial resolution. A $250~\text{mJ}~\text{pulse}^{-1}$ Nd:YAG laser (BMI 5000) was used for illumination. The scattered light from particles was captured by four CCD cameras with $2048~\text{pixel}\times 2048~\text{pixel}$ (Hamamatsu C9300-024) through Nikon 105 mm lenses. The f #8 aperture gave a diffraction spot of approximately 2 pixels. These cameras were mounted in Scheimpflug conditions (Willert Reference Willert1997) and the viewing angle and distance between two cameras were $45^{\circ }$ and 1.37 m respectively. For seeding, polyethylene glycol particles with a diameter of $1~{\rm\mu}\text{m}$ were generated by a smoke generator. Each stereo PIV plane had $16~\text{cm}\times 11~\text{cm}$ field of view, and the combined field of $31~\text{cm}\times 11~\text{cm}$ was obtained with a small overlap. The images from the four cameras were acquired by two frame grabbers (X64 Xcelera-CL PX4), and digitized images were recorded on PC hard drives. The sampling rate of stereo PIV was 4 Hz.

The streamwise position of the light sheet was adjusted to be aligned with the centre of the holes of the pressure probe and the wall pressure tap. The two light sheets were slightly separated in the $x$ direction (approximately 0.75 mm) to obtain better correlation; it should be noted that the light sheet thickness was approximately 1.85 mm. The angle of the light sheet was carefully adjusted around both $y$ and $z$ axes. The time separation between two exposures was optimized: ${\rm\Delta}t=250~{\rm\mu}\text{s},150~{\rm\mu}\text{s}$ and $75~{\rm\mu}\text{s}$ for $3~\text{m}~\text{s}^{-1},5~\text{m}~\text{s}^{-1}$ and $10~\text{m}~\text{s}^{-1}$ respectively in order to get a maximum displacement of the order of 10 pixels.

The PIV analysis was performed by a standard multi-pass FFT-based cross-correlation method with integer shift of both windows (Lin et al. Reference Lin, Foucaut, Laval, Pérenne and Stanislas2008; Herpin et al. Reference Herpin, Stanislas, Foucaut and Coudert2013). A 1D Gaussian peak fitting algorithm was used for the sub-pixel displacement determination. Three passes with different window sizes were used. The interrogation window size of the final pass was $26~\text{pixel}\times 39~\text{pixel}$ . The scale relationship between physical and image spaces was $11.6~\text{pixel}~\text{mm}^{-1}$ in $y$ and $17.6~\text{pixel}~\text{mm}^{-1}$ in $z$ . Therefore, the physical size of the interrogation window ( ${\it\Delta}_{y}$ and ${\it\Delta}_{z}$ in $y$ and $z$ directions) was $2.24~\text{mm}\times 2.21~\text{mm}$ , corresponding to about 17–50 wall units at $\mathit{Re}_{{\it\theta}}=7300$ and 18 000 respectively. The velocity was computed at grid points within the ranges $1~\text{mm}\leqslant y\leqslant 308~\text{mm}$ and $-0.5~\text{mm}\leqslant z\leqslant 110~\text{mm}$ . The numbers of grid points in $y$ and $z$ were 615 and 222 respectively for the combined field, and 45 points were overlapped in the $y$ direction. The PIV overlap ratio was 77 %. These PIV parameters such as the measurement area and interrogation window size relative to the representative length scale of the flow are summarized in table 4. The Soloff method (Soloff, Adrian & Liu Reference Soloff, Adrian and Liu1997) was used to reconstruct the three velocity components. From a set of calibration and stereo PIV particle images, the misalignment between the light sheet and the calibration plane was compensated for using the technique described in Coudert & Schon (Reference Coudert and Schon2001).

2.3. Simultaneous measurement of fluctuating pressure and velocity fields

Each run of simultaneous measurement was repeated for nine different wall-normal positions of the pressure probe given in table 5. The number of valid realizations of the velocity field is 10 000 for each run. This corresponds to pressure recordings spanning 2500 s at each position and at a sampling rate of 40 kHz.

One important question with such an experimental set-up is the perturbation generated by the pressure probe very close to the wall, although great care was taken to minimize its size and intrusion at wall distances comparable to its diameter. The tip of the probe was 19.5 mm upstream of the wall sensor. This was carefully checked as 10 000 samples of wall pressure and velocity fields were available for each probe position. If the probe has no influence on the wall pressure or on the near-wall flow field, no difference should be seen between these nine packets of 10 000 samples. To achieve this, the wall pressure–velocity correlations were compared between the different packets. This was done for all cases and the result, not shown here, is that only the streamwise velocity–wall pressure correlation shows a visible effect, only for probe positions (b–d). The effect observed is that the correlation shape is not changed, only the values of the correlation coefficient are slightly enhanced. Our interpretation of this result is that when the probe is very close to the wall, it tends to stabilize the meandering wall streaks of the inner layer spatially, thus enhancing the correlation observed.

For this reason, the $\boldsymbol{R}_{pu}$ streamwise velocity–wall pressure correlations defined later will be averaged on 60 000 samples only (corresponding to probe positions (e–j)), while the two other correlations ( $\boldsymbol{R}_{pv}$ and $\boldsymbol{R}_{pw}$ ) will be averaged on 90 000 samples as they show no probe influence.

For the wall pressure fluctuations, the random error on r.m.s. values, with the sampling time of 600 s used, is evaluated to be $\pm 1.3\,\%$ . A slight bias due to the probe proximity is visible in the wall pressure fluctuations for the probe positions closer than $y_{p}=10~\text{mm}$ , and it is at most $\pm 2.5\,\%$ . It is expected that the pressure fluctuations in the field measured by the probe will have a similar tendency for the uncertainty since the instrumentation and procedure are the same except for the difference in the probe geometry. For the velocity data, the bias of each stereo PIV system can be estimated from the overlapped region of two PIV planes. On average, the bias is at most $0.015~\text{m}~\text{s}^{-1}$ for the $U_{e}=5~\text{m}~\text{s}^{-1}$ case. The random error in the r.m.s. value of the velocity fluctuations with 10 000 samples is estimated to be $\pm 2{-}3\,\%$ in the logarithmic region. For the pressure–velocity correlations, those with the wall pressure allow convergence to be analysed as 90 000 samples are available. This analysis is not presented here. Although the correlation contours are less converged with 10 000 samples compared with the ones from 90 000 samples, this convergence appears clearly enough to evaluate the global extension of the wall pressure–velocity correlation for all three components. It is considered here that the field pressure–velocity correlations have the same level of convergence as the wall pressure–velocity correlations when both are computed from 10 000 samples.

Figure 8. Comparison of stereo PIV and hot-wire. (a) Mean velocity profiles $U^{+}$ . The log-law constants ${\it\kappa}$ and $B$ are 0.41 and 5.2 respectively. (b) The r.m.s. of the velocity fluctuations $u^{\prime +}$ (black), $v^{\prime +}$ (grey) and $w^{\prime +}$ (light grey).

4.1. Wall pressure–velocity correlations

As was mentioned earlier, with the fixed point at the wall, the correlations converge better when they are averaged on 60 000 ( $\boldsymbol{R}_{pu}$ ) or 90 000 ( $\boldsymbol{R}_{pv}$ and $\boldsymbol{R}_{pw}$ ) samples, which is helpful for analysis. Moreover, for interpretation, it is of interest to split the wall pressure fluctuations into two parts: the positive ones which can be associated with a decrease of the wall-parallel (streamwise or spanwise) velocity component (Bernoulli static pressure effect) or an increase of the wall-normal component (stagnation point effect) with respect to the mean. It is in fact just the opposite for negative pressure fluctuations.

Figure 14 gives the $\boldsymbol{R}_{pu}$ correlation in the ${\rm\Delta}t{-}y$ plane for $z/{\it\delta}=0$ for the three Reynolds numbers. In order to estimate the size of the different correlation regions, here and in the following cuts, a thin black contour line is plotted, corresponding to a correlation level of $\pm 3\times 10^{-6}$ , which is slightly above the noise level. Thanks to spanwise homogeneity, it is expected that this correlation will be symmetric with respect to the $z=0$ plane. As can be observed, it is mostly positive on the negative ${\rm\Delta}t$ side and negative on the other side. The time extension and the size along $y$ grow significantly with Reynolds number. This is true along $z$ as well. The maximum wall-normal extension is of order $0.5{\it\delta}$ , $0.6{\it\delta}$ and $0.8{\it\delta}$ for $\mathit{Re}_{{\it\theta}}=7300$ , 10 000 and 18 000 respectively. Taking into account the symmetry with respect to $z/{\it\delta}=0$ (due to spanwise homogeneity), the spanwise extent grows up to about ${\it\delta}~(\pm {\it\delta}/2)$ at the highest Reynolds number. It is interesting to note that the pressure fluctuations appear located at the leading edge of this large positive ‘structure’. At ${\rm\Delta}t=0$ , the maximum of the correlation is slightly above the wall, in the buffer layer. The spanwise extent at $\mathit{Re}_{{\it\theta}}=10\,000$ is approximately $\pm 0.2{\it\delta}$ at ${\rm\Delta}t=0$ , but it increases beyond $\pm 0.4{\it\delta}$ when going upstream (negative ${\rm\Delta}t$ ). The same is true for the wall-normal extension which is maximum at approximately ${\rm\Delta}tU_{e}/{\it\delta}\simeq -0.4/-0.6$ . The main effect of the Reynolds number is to increase the size of the positive correlation region in all directions, which means that it does not scale in external variables.

Figure 14. Cuts of the wall pressure $\boldsymbol{R}_{pu}$ (case (a) of table 5) in the ${\rm\Delta}t{-}y$ plane at $z=0$ at $\mathit{Re}_{{\it\theta}}=7300$  (a), 10 000 (b) and 18 000 (c).

To refine the analysis, it is of interest to split the correlation between positive and negative pressure fluctuations by conditional averaging. This is done for the intermediate Reynolds number (which is representative of the two others) and plotted in the plane $z=0$ in figure 15(a,b) for each sign respectively. In order to get a better view of the 3D structure of these two conditional correlations, two movies are provided as extra electronic supplementary movies available at http://dx.doi.org/10.1017/jfm.2015.158 (movies 1 and 2). The captions of the movies are listed in appendix A.

Figure 15. Cuts of the wall pressure $\boldsymbol{R}_{pu}$ (case (a) of table 5) in the ${\rm\Delta}t{-}y$ plane at $z/{\it\delta}=0$ and at $\mathit{Re}_{{\it\theta}}=10\,000$ with (a)  $p>0$ and (b)  $p<0$ .

For the positive pressure fluctuations (figure 15 a), two main facts can be noted. (i) The intensity of the positive correlation region, corresponding to positive $u$ (i.e. high speed), increases and extends further upstream and downstream down to $0.5{\it\delta}/U_{e}$ . A weak elongated negative correlation region appears on the sides of this large upstream positive region extending outside the field of view in span. The spanwise and wall-normal extension of the positive correlation region does not change significantly. (ii) A significant region of negative correlation appears on the positive ${\rm\Delta}t$ side, strongly inclined along the wall and attached to it. Plotting it for larger ${\rm\Delta}t$ shows that it extends down to about $10{\it\delta}/U_{e}$ where it occupies nearly the full boundary layer thickness. It corresponds to negative velocity fluctuations, i.e. to a low-speed region. The only coherent low-speed regions that have been identified previously as extending from the wall up in the boundary layer with such a shape are the hairpin packets of Adrian et al. (Reference Adrian, Meinhart and Tomkins2000), though not on such time extension.

The negative pressure fluctuation correlation is shown in figure 15(b). The shape is completely different. The upstream large positive correlation region disappears completely and is replaced by a very small and intense region very close to the wall and to the fixed point. The dominating feature is a large negative correlation region on the positive ${\rm\Delta}t$ side (extending nearly to ${\it\delta}$ in $y$ , $2.5{\it\delta}/U_{e}$ in time and with a spanwise extent of the order of ${\it\delta}$ ). This region corresponds to high speeds since the pressure fluctuation is negative.

From these figures and movies, it is clear that positive and negative pressure fluctuations at the wall are associated with very different streamwise velocity fluctuation organizations above them. It is thus of interest to look at the correlation with the other fluctuating velocity components, available here thanks to the stereoscopic PIV (SPIV) set-up used.

Figure 16 gives the $\boldsymbol{R}_{pv}$ correlation in the ${\rm\Delta}t{-}y$ plane for $z/{\it\delta}=0$ for the three Reynolds numbers. As can be observed, this correlation is relatively localized in time but, as a difference from $\boldsymbol{R}_{pu}$ , it is closer to antisymmetry with respect to the ${\rm\Delta}t=0$ axis. The extent is of order 1 in both $y/{\it\delta}$ and ${\rm\Delta}tU_{e}/{\it\delta}$ , with a maximum of each lobe quite close to the fixed point. It is noticeable that, as a difference from $\boldsymbol{R}_{pu}$ , the Reynolds number seems to have little effect on this correlation. In the $z=0$ plane of figure 16, both the positive and the negative lobes are at a strong angle to the wall (close to $45^{\circ }$ ).

Figure 16. Cuts of the wall pressure $\boldsymbol{R}_{pv}$ (case (a) of table 5) in the ${\rm\Delta}t{-}y$ plane at $z=0$ at $\mathit{Re}_{{\it\theta}}=7300$  (a), 10 000 (b) and 18 000 (c).

Here again, it is of interest to split the correlation between positive and negative pressure fluctuations by conditional averaging. This is done for the intermediate Reynolds number and is plotted in the $z=0$ plane in figure 17(a,b) for each sign respectively. Furthermore, in order to get a better view of the 3D structure of these two conditional correlations, two movies are provided as extra electronic material (movies 3 and 4).

Figure 17. Cuts of the wall pressure $\boldsymbol{R}_{pv}$ (case (a) of table 5) in the ${\rm\Delta}t{-}y$ plane at $z/{\it\delta}=0$ and at $\mathit{Re}_{{\it\theta}}=10\,000$ with (a)  $p>0$ and (b)  $p<0$ .

The $p>0$ correlation of figure 17(a) is dominated by the upstream negative correlation region. The global shape of this correlation is indicative of a strong downward motion upstream of the positive pressure event immediately followed by a weak outward motion downstream, both separated by a sharp interface above the fixed point.

On the contrary, the $p<0$ correlation is dominated by the positive correlation region downstream. There is still a significant negative correlation very close to the fixed point and surrounding it. As the pressure fluctuation is now negative, the downstream motion is wallward and the upstream one is outward, just opposite to the previous case, but again with the pressure event just at the interface. As can be seen, the wall-normal velocity fluctuation organization is again significantly different between positive and negative pressure fluctuations at the wall. It should also be noted that the time extension of these correlations is much less than for $\boldsymbol{R}_{pu}$ . No near-wall elongated structure is visible.

To complete the picture, it is of interest to look at $\boldsymbol{R}_{pw}$ . For homogeneity reasons, this correlation is zero in the $z/{\it\delta}=0$ plane and antisymmetric with respect to this plane. Figure 18 gives $\boldsymbol{R}_{pw}$ in the ${\rm\Delta}t{-}y$ plane for $z/{\it\delta}=0.2$ and for the three Reynolds numbers. This correlation is obviously the most complex of the three. Its spanwise extent is larger than the field of view (which is $0.4{\it\delta}$ ). It is strongly inclined to the wall along $z$ ( $20^{\circ }{-}25^{\circ }$ ) and this inclination increases with Reynolds number. It is quite extensive in the wall-normal direction (up to $y/{\it\delta}=0.6$ ), but also in time, on both positive and negative ${\rm\Delta}t$ sides ( $\pm {\it\delta}/U_{e}$ at the highest Reynolds number). Significant negative correlation regions are observed both downstream of the fixed point (positive ${\rm\Delta}t$ ) close to the wall and upstream (negative ${\rm\Delta}t$ ) further away from the wall. Overall, this leads to fairly large $w$ motions involved in the wall pressure fluctuations. On increasing the Reynolds number, the positive region strengthens (especially between 7300 and 10 000). The negative downstream region is squeezed against the wall and reduced in intensity. The upstream negative region seems to grow slowly in size and intensity.

Figure 18. Cuts of the wall pressure $\boldsymbol{R}_{pw}$ (case (a) of table 5) in the ${\rm\Delta}t{-}y$ plane at $z/{\it\delta}=0.2$ at $\mathit{Re}_{{\it\theta}}=7300$  (a), 10 000 (b) and 18 000 (c).

In figure 19(a,b), the correlation is again split between positive and negative pressure fluctuations, for $\mathit{Re}_{{\it\theta}}=10\,000$ and for each sign respectively. In order to get a better view of the 3D structure of these two conditional correlations, two films are provided as extra electronic material (movies 5 and 6).

Figure 19. Cuts of the wall pressure $\boldsymbol{R}_{pw}$ (case (a) of table 5) in the ${\rm\Delta}t{-}y$ plane at $z/{\it\delta}=0.2$ and at $\mathit{Re}_{{\it\theta}}=10\,000$ with (a)  $p>0$ and (b)  $p<0$ .

For $p>0$ , the downstream negative correlation nearly disappears. For $p<0$ it is the upstream one that disappears while the downstream one strengthens significantly. The positive region around the fixed point is also quite affected by conditioning. In both cases, the spanwise extension is much wider than the field of view and, as for $\boldsymbol{R}_{pv}$ , the time extension of the correlation is limited compared with $\boldsymbol{R}_{pu}$ . In fact, these correlations show significant spanwise inward and outward motions at the scale of the boundary layer, coupled in a way that is indicative of large more or less streamwise vortical motions.

Looking at the different plots of the three pressure–velocity space–time correlations presented in figures 14–19 and looking also at movies 1–6, it is clear that the wall pressure is coupled to various coherent flow structures which occupy a large region of the boundary layer. Overall, these pressure fluctuations are sensitive to velocity events inside a box that is about ${\it\delta}$ in $y$ , ${\it\delta}$ in $z$ (taking into account the symmetries) and more than $[-4{\it\delta}/U_{e},10{\it\delta}/U_{e}]$ in time. These highly correlated velocity events are in fact combinations of the three velocity components which change significantly depending on the pressure fluctuation sign. Figures 20 and 21 try to make a synthesis for each pressure fluctuation sign at $\mathit{Re}_{{\it\theta}}=10\,000$ . In each figure, the correlation has been thresholded at a level allowing the main positive and negative regions of each to be highlighted. By looking at movies 1–6, a more complete understanding of the correlation shape is made possible. As it is not easy to understand the full flow structure associated with each pressure event from these separate correlations, two extra movies have been made which combine the three correlations $\boldsymbol{R}_{pu},\boldsymbol{R}_{pv}$ and $\boldsymbol{R}_{pw}$ in a single 3D representation. The image is presented in the $y{-}z$ wall-normal–spanwise plane as a function of time. The $\boldsymbol{R}_{pv}$ and $\boldsymbol{R}_{pw}$ correlations are represented as vectors in the plane and $\boldsymbol{R}_{pu}$ is coded in colour as an out-of-plane component. The two films scan in time through the space–time pressure–‘velocity vector’ correlation. Movies 7 and 8 give the results for positive and negative pressure fluctuations at the wall respectively. Some selected cuts from these two movies are plotted in figures 22 and 23.

Figure 20. Three-dimensional plots of thresholded $\boldsymbol{R}_{pu}$  (a),  $\boldsymbol{R}_{pv}$  (b) and $\boldsymbol{R}_{pw}$  (c) correlation for $p>0$ at the wall at $\mathit{Re}_{{\it\theta}}=10\,000$ . Contour levels are $-7.5\times 10^{-6}$ (light blue), $1.5\times 10^{-5}$ (orange) for $\boldsymbol{R}_{pu}$ , $-7.5\times 10^{-6}$ (light blue), $6\times 10^{-6}$ (orange) for $\boldsymbol{R}_{pv}$ and $-7.5\times 10^{-6}$ (light blue), $7.5\times 10^{-6}$ (orange) for $\boldsymbol{R}_{pw}$ .

Figure 21. Three-dimensional plots of thresholded $\boldsymbol{R}_{pu}$  (a),  $\boldsymbol{R}_{pv}$  (b) and $\boldsymbol{R}_{pw}$  (c) correlation for $p<0$ at the wall at $\mathit{Re}_{{\it\theta}}=10\,000$ . Contour levels are $-1.5\times 10^{-5}$ (light blue), $9\times 10^{-6}$ (orange) for $\boldsymbol{R}_{pu}$ , $-7.5\times 10^{-6}$ (light blue), $7.5\times 10^{-6}$ (orange) for $\boldsymbol{R}_{pv}$ and $-7.5\times 10^{-6}$ (light blue), $7.5\times 10^{-6}$ (orange) for $\boldsymbol{R}_{pw}$ .

Figure 22. Cuts in the $y{-}z$ plane of space–time ‘wall pressure’–‘velocity vector’ correlations for $p>0$ : (a)  ${\rm\Delta}tU_{e}/{\it\delta}=-1.46$ ; (b)  ${\rm\Delta}tU_{e}/{\it\delta}=-0.46$ ; (c)  ${\rm\Delta}tU_{e}/{\it\delta}=-0.00$ ; (d)  ${\rm\Delta}tU_{e}/{\it\delta}=0.16$ ; (e)  ${\rm\Delta}tU_{e}/{\it\delta}=0.36$ ; (f)  ${\rm\Delta}tU_{e}/{\it\delta}=0.96$ .

Figure 23. Cuts in the $y{-}z$ plane of space–time ‘wall pressure’–‘velocity vector’ correlations for $p<0$ : (a)  ${\rm\Delta}tU_{e}/{\it\delta}=-0.26$ ; (b)  ${\rm\Delta}tU_{e}/{\it\delta}=-0.06$ ; (c)  ${\rm\Delta}tU_{e}/{\it\delta}=0.06$ ; (d)  ${\rm\Delta}tU_{e}/{\it\delta}=0.26$ ; (e)  ${\rm\Delta}tU_{e}/{\it\delta}=0.46$ ; (f)  ${\rm\Delta}tU_{e}/{\it\delta}=0.86$ . The signs of the correlation are flipped to represent the flow pattern associated with the negative wall pressure.

Looking first at the positive pressure fluctuations at the wall, the different correlations plotted in figures 20, 22 and movie 7 give a clear overall picture of a strong sweeping event coming from upstream. Clear traces of streamwise vortices are detectable on both sides, along a line at an angle of $\pm 45^{\circ }$ spanwise. This leads to the conclusion that this positive $p$ is mostly due to a total pressure effect linked to the $v$ component. It is also clear from movie 1 and figure 14 that the large upstream positive correlation region is flanked on both sides by elongated negative correlations close to the wall and extending out of the field of view in span.

For the negative pressure fluctuations at the wall illustrated in figures 21 and 23 and movie 8 (to take into account the fact that $p<0$ , the signs of the correlations have been changed to make the movie), low-speed ejection moves down to the wall, pushing the near-wall high streamwise velocity region on the sides and focusing close to the fixed point. It is immediately followed, for ${\rm\Delta}t>0$ by a strong sweeping motion focalized at the origin but involving the whole boundary layer thickness. Evidence of streamwise vortices is observable in the nearby low-streamwise-velocity regions. These streamwise vortices, which are relatively large, seem to lift up progressively on increasing ${\rm\Delta}t$ .

As can be seen, both positive and negative pressure fluctuations at the wall seem to appear at sharp interfaces of very large organized motions at the scale of the boundary layer thickness. In both cases, the pressure fluctuation seems to be mostly due to strong variations of the wall-normal velocity. It is also important to notice that the large structures appearing upstream for $p>0$ and downstream for $p<0$ are of the same nature: a strong sweeping motion, involving the whole boundary layer thickness and having a time extension of the order of ${\it\delta}/U_{e}$ . These sweeping motions are flanked on both sides by streamwise vortical motions of order $0.2{\it\delta}$ in diameter, which are much larger than the usual near-wall streamwise vortices (which are approximately 20 wall units in diameter).

4.2. Field pressure–velocity correlations

Having analysed in detail the pressure–velocity correlations with the fixed pressure point at the wall, it is of interest to look now at what happens in the field since it appeared to be quite different in figure 12.

Looking first at $\boldsymbol{R}_{pu}$ , as shown in figure 13, a clear Reynolds number effect is visible on this correlation. It concerns essentially the size, as the overall characteristics of the correlation are similar for the three different Reynolds numbers. To look at the correlation structure, it is interesting to complement figure 12 by cuts in the $z=0$ plane, which is a plane of symmetry of this correlation. This is done in figure 24 at $\mathit{Re}_{{\it\theta}}=10\,000$ , for positions (b–j) of the pressure point as given in table 5, and in figure 25 for position (b) and for the three Reynolds numbers investigated.

Figure 24. Cuts of the field pressure $\boldsymbol{R}_{pu}$ in the ${\rm\Delta}t{-}y$ plane at $z=0$ and at $\mathit{Re}_{{\it\theta}}=10\,000$ . The dotted line indicates the pressure reference position as given in table 5.

Figure 25. Cuts of the field pressure $\boldsymbol{R}_{pu}$ correlation in the ${\rm\Delta}t{-}y$ plane at $z=0$ for three Reynolds numbers (increasing from (a) to (c)) at the reference position (b) in table 5.

At first glance, differently from the pressure point at the wall, $\boldsymbol{R}_{pu}$ exhibits an inclined elongated shape on the positive ${\rm\Delta}t$ side. At the highest Reynolds number, the near-wall negative lobe extends beyond the limit of the field of view on the positive ${\rm\Delta}t$ side (up to ${\rm\Delta}tU_{e}/{\it\delta}\sim 18$ at the highest Reynolds number). Both of these positive and negative correlation regions are inclined at a small angle with respect to the ${\rm\Delta}t$ axis. This angle decreases when the Reynolds number increases ( $7^{\circ },5^{\circ }$ and $3^{\circ }$ respectively at the three Reynolds numbers for the positive part). Obviously linked to the main elongated positive region, it is present for all Reynolds numbers with comparable size, position and intensity and it extends beyond $z/{\it\delta}=0.4$ .

Looking at the main positive correlation region, for wall distances smaller than $y_{p}/{\it\delta}\sim 0.15$ (b–f), i.e. in the inner part of the boundary layer, two characteristic patterns are observed: a ‘smaller’ one, which is relatively localized in negative ${\rm\Delta}t$ and around the fixed point, and a longer one, which is inclined and elongated in the positive ${\rm\Delta}t$ direction with a shallow angle to the horizontal axis. The small region on the negative ${\rm\Delta}t$ side is obviously related with the correlation observed for the fixed point at the wall. In this inner region, for a given Reynolds number, the shape and size of the positive correlation region change little with $y$ . Beyond $y_{p}/{\it\delta}\sim 0.1$ , the two lobes of the positive correlation region merge and form one large correlation pattern. Clearly, the elongated part of the correlation is a wall-attached structure subjected to stretching by convection, which is also supported by the upstream displacement of the structure when moving the fixed point away from the wall. This could be the wall-attached eddies of Townsend.

In order to assess the global shape of this correlation in the inner layer, a movie is provided as extra electronic material for position (b) of table 5 (movie 9). The weak negative correlation on the side of the positive one up to (h), extending significantly outside of the field of view in $z$ , is clearly evidenced by this movie, as well as the elongated negative correlation region on the downstream side, extending far outside the field of view, down to ${\rm\Delta}tU_{e}/{\it\delta}\sim 18$ .

The second correlation of interest is $\boldsymbol{R}_{pv}$ . A cut in the ${\rm\Delta}t{-}y$ plane at $z=0$ is given in figure 26 at $\mathit{Re}_{{\it\theta}}=10\,000$ , which is again highly representative of the shape of this correlation at the three Reynolds numbers. For the purpose of comparison, the correlation with the wall pressure at point (a) is also shown in this figure. As the wall-normal velocity fluctuations are known to be smaller than the streamwise ones the correlation is much weaker than $\boldsymbol{R}_{pu}$ (colour scale range is half).

Figure 26. Cuts of the field pressure $\boldsymbol{R}_{pv}$ in the ${\rm\Delta}t{-}y$ plane at $z=0$ and at $\mathit{Re}_{{\it\theta}}=10\,000$ . The dotted line indicates the pressure reference position as given in table 5.

From cuts in different planes, it appears that the size and shape of the lobes are very similar in the whole inner region. Taking into account the symmetry with respect to the $z=0$ plane, they are approximately $0.4{\it\delta}$ in span and $0.6{\it\delta}$ in wall-normal extent. In the outer part, the temporal extent of the negative lobe is of the order of $4{\it\delta}/U_{e}$ . From plots in the same planes at the two other Reynolds numbers (not shown here), it appears that the effect of this parameter on the size and shape of this correlation is much weaker than for $\boldsymbol{R}_{pu}$ .

The last correlation to be analysed is $\boldsymbol{R}_{pw}$ . An example is plotted in 3D perspective in figure 27 for pressure reference position (b) at the intermediate Reynolds number $\mathit{Re}_{{\it\theta}}=10\,000$ . This is the most complex correlation of the three as it has several distinct regions of each sign and shows significant changes with both wall distance and Reynolds number. Two movies are provided as extra electronic material to show the 3D structure at $\mathit{Re}_{{\it\theta}}=10\,000$ in the inner (b) (movie 10) and outer (h) (movie 11) regions respectively. As far as the Reynolds number effect is concerned, it does not greatly change the global picture presented above. The shape of this outer region structure coupling a positive and negative correlation region closely resembles an elongated streamwise oriented vortical structure with a size of the order of the boundary layer thickness. It is interesting to note that, in the pressure creating events, a positive $w$ motion in one region is always associated with a negative one above or below it, and vice versa, which is indicative of streamwise vortical motions.

Figure 27. Three-dimensional representation of $\boldsymbol{R}_{pw}$ at $\mathit{Re}_{{\it\theta}}=10\,000$ for pressure reference position (b) as given in table 5. Contour levels are $-8\times 10^{-6}$ (light blue), $8\times 10^{-6}$ (orange) and $1.6\times 10^{-5}$ (red).

To complete this analysis and in the context of the ongoing research around large-scale structures in turbulent boundary layers, it is of interest to emphasize some specific Reynolds number effects. The extension and location of $\boldsymbol{R}_{pu}$ changes significantly in the wall-normal direction, while the spanwise extension is nearly the same. This wallward extension is accompanied by a downward motion of $\boldsymbol{R}_{pw}$ .

This strong Reynolds number influence is visible in figure 28, which shows $\boldsymbol{R}_{pu}$ in the $z=0$ plane at point (i) corresponding to $y_{p}/{\it\delta}\simeq 0.8$ at the three Reynolds numbers.

Figure 28. Cuts of the field pressure $\boldsymbol{R}_{pu}$ correlation in the ${\rm\Delta}t{-}y$ plane at $z=0$ , at the pressure reference position (i) in table 5 and for different Reynolds numbers (increasing from (a) to (c)).

As for the wall pressure correlations, in order to assess the global flow structure associated with the correlation shapes, it is of interest to make movies combining the three correlations in a single representation. The correlation has been here again conditioned on the sign of the local fluctuating pressure. What happens close to the wall in the inner layer is focused on. Point (c) located at $y^{+}=56$ , i.e. in the buffer layer, is selected for this. Movie 12 corresponds to $p>0$ and movie 13 to $p<0$ . Specific cuts in the $y$ – $z$ plane are given at different ${\rm\Delta}t$ in figures 29 and 30.

Figure 29. Cuts in the $y{-}z$ plane of space–time ‘field pressure’–‘velocity vector’ correlations for $p>0$ : (a)  ${\rm\Delta}tU_{e}/{\it\delta}=0.00$ ; (b)  ${\rm\Delta}tU_{e}/{\it\delta}=0.35$ ; (c)  ${\rm\Delta}tU_{e}/{\it\delta}=0.99$ ; (d)  ${\rm\Delta}tU_{e}/{\it\delta}=3.70$ ; (e)  ${\rm\Delta}tU_{e}/{\it\delta}=6.02$ ; (f)  ${\rm\Delta}tU_{e}/{\it\delta}=8.01$ . The probe position is position (c) in table 5.

Figure 30. Cuts in the $y{-}z$ plane of space–time ‘field pressure’–‘velocity vector’ correlations for $p<0$ : (a)  ${\rm\Delta}tU_{e}/{\it\delta}=-0.57$ ; (b)  ${\rm\Delta}tU_{e}/{\it\delta}=-0.07$ ; (c)  ${\rm\Delta}tU_{e}/{\it\delta}=0.07$ ; (d)  ${\rm\Delta}tU_{e}/{\it\delta}=0.86$ ; (e)  ${\rm\Delta}tU_{e}/{\it\delta}=3.03$ ; (f)  ${\rm\Delta}tU_{e}/{\it\delta}=5.04$ . The probe position is position (c) in table 5. The signs of the correlation are flipped to represent the flow pattern associated with the negative wall pressure.

Looking first at positive pressure fluctuations (movie 12 and figure 29), the flow structure observed for negative ${\rm\Delta}t$ is completely similar to the one appearing in movie 7 and figure 22 for positive wall pressure fluctuations. This explains why no cuts are provided in figure 29 which focuses on the positive ${\rm\Delta}t$ part. In this region, the flow structure observed is very different from that obtained with the wall pressure probe. At ${\rm\Delta}tU_{e}/{\it\delta}=0.35$ , although an outward motion is visible in the outer part, a strong high-streamwise-velocity region is still present in the inner layer with clear traces of streamwise vortices and downward motion due to them. This high-speed region extends progressively outward, flanked by the vortices which move progressively outwards too. At ${\rm\Delta}tU_{e}/{\it\delta}=3.7$ , the high-speed region influences the whole boundary layer thickness, the vortices are localized around $0.35{\it\delta}$ and a global sweeping motion is induced between them. A low-speed region starts to appear at the wall. Downstream, the high-speed region fades away while the low-speed region develops and occupies approximately ${\it\delta}/2$ at ${\rm\Delta}tU_{e}/{\it\delta}=8$ . Some vortical motions and weak ejections are detectable inside this low-speed region but these are much less coherent than those associated with the preceding high-speed region. It should be remembered that the symmetry of the correlations is enforced here by the spanwise homogeneity, which means that the counter-rotating vortex pairs observed are statistical and can be the result of many single vortices of the same sign. It is clear nevertheless that a streamwise vortex on the side of a high-speed region is rotating in order to contribute to downward motion of the fluid. What is also obvious is that on average, these streamwise vortical motions have a size of the order of $0.1{\it\delta}$ , which is much larger than the wall generated vortices found in this type of flow (Tanahashi et al. Reference Tanahashi, Kang, Miyamoto, Shiokawa and Miyauchi2004; Herpin et al. Reference Herpin, Stanislas, Foucaut and Coudert2013). Finally, it seems that the flow structure sensed by the field pressure on the positive ${\rm\Delta}t$ is not directly related to the one at the wall which, looking at movie 7 and figure 22, shows a very quick fading of the high-speed region after ${\rm\Delta}t=0$ and the development of a low-speed near-wall region at ${\rm\Delta}tU_{e}/{\it\delta}=0.35$ .

The negative pressure fluctuation case is described by movie 13 and figure 30. Here, the near-wall region is dominated by a low-speed elongated structure which appears at approximately ${\rm\Delta}tU_{e}/{\it\delta}=-0.6$ and disappears at approximately ${\rm\Delta}tU_{e}/{\it\delta}=5$ . This low-speed region, which is originally of weak ejection type, is perturbed just after ${\rm\Delta}t=0$ by a strong and focalized sweeping motion which is sensed also by the wall pressure (compare figures 23 and 30 at ${\rm\Delta}tU_{e}/{\it\delta}=0.06$ ). However, as a difference here, the low-speed region is not suppressed, it redevelops under the high-speed sweep which moves away from the wall and evidences a strong ejection fed by strong near-wall vortical structures at the scale of the inner layer ( $0.1{-}0.2{\it\delta}$ ). Both the low-speed region and the vortical structures stay attached to the wall down to approximately ${\rm\Delta}tU_{e}/{\it\delta}=1.5$ . The outer high-speed sweeping region disappears around this time, allowing both the low-speed region and the companion vortices to lift progressively away from the wall and to disappear at approximately ${\rm\Delta}tU_{e}/{\it\delta}=5$ . A near-wall high-speed region develops from ${\rm\Delta}tU_{e}/{\it\delta}=4$ , but with no clear evidence of vortices or ejection.

As can be seen, apart from the strong sweep after ${\rm\Delta}t=0$ , this second case is quite different from what is correlated with the wall pressure.