2.1 Wind tunnel

The experiments reported in this article were performed in the TFT boundary-layer wind tunnel, which was specifically designed and fabricated for this purpose (Mohammed-Taifour et al. Reference Mohammed-Taifour, Schwaab, Pioton and Weiss2015a ). This is a low-speed, blow-down wind tunnel with a 3 m long and 0.6 m wide test section. The test surface along which the boundary layer develops, separates and reattaches is the top surface of the test section. In its first half, a nominally two-dimensional, turbulent boundary layer develops in a zero-pressure-gradient (ZPG) zone. In its second half, the boundary layer separates from the test surface because of a strong adverse pressure gradient and reattaches further downstream thanks to a favourable pressure gradient.

Figure 1 shows a side view sketch of the test section, as well as the pressure distribution on the top surface along the longitudinal axis for three positions across the span. Like all other data presented in this article, these measurements were obtained at a reference velocity of $U_{ref}=25~\text{m}~\text{s}^{-1}$ , where the reference position is located at the midpoint of the test-section entrance plane. The origin of the axis system is also at the entrance of the test section, but on the centreline of the top surface: $x$ is the longitudinal direction, $y$ is the wall-normal direction (positive going down) and $z$ is the spanwise direction (positive towards the paper).

Figure 1.

Test section sketch and pressure distribution in the TFT boundary-layer wind tunnel, $U_{ref}=25~\text{m}~\text{s}^{-1}$ . Black and red rectangles and numbers indicate PIV measurement stations (§ 2.2).

The blue dashed line in figure 1 represents the 99 % boundary-layer thickness on the test-surface centreline, which was obtained from vertical profiles measured with a hot-wire anemometer. The boundary layer developing on the lower surface (not represented in figure 1) is removed by a bleed slot in the upstream part of the pressure-gradient zone. This slot connects directly to the atmosphere, while the interior of the test section is maintained at a slightly elevated pressure by a mesh positioned at the exit. This system maintains the TSB on the flat ceiling of the test section and it was verified that no separation occurs on the lower contoured wall.

The pressure coefficient upstream of $x=1.5$  m is essentially zero, which is achieved by a slight divergence ( $0.4^{\circ }$ ) of the lower surface. At $x=1.5$  m, the floor diverges abruptly, which strongly increases the pressure coefficient. $c_{p}$ then increases more slowly between $x=1.75$  m and $x=2.25$  m before decreasing abruptly due to the floor convergence. For $x>2.5$  m, the pressure coefficient remains approximately constant at a slightly elevated value compared to the ZPG zone. The pressure distribution in the test section generates a massively separated TSB on the top surface, which is sketched in figure 1 by the dividing streamline shown as a blue solid line. Note that the $c_{p}$ distribution in the pressure-gradient zone is very similar to those generated in the experiment of Patrick (Reference Patrick1987) and the DNS of Na & Moin (Reference Na and Moin1998a ).

Figure 2 shows the result of oil-film visualisation experiments on the top surface and on one side wall of the test section in the region of the TSB. The coating was a mixture of titanium dioxide, paraffin oil, and some oleic acid. It was applied on the surfaces before turning on the wind tunnel. Note that the exact proportions of the mixture were slightly different for both surfaces and that the corresponding images were not taken simultaneously. The critical points of the shear-stress field and the limiting streamlines connecting them were drawn on the image to facilitate its interpretation.

Figure 2.

Oil-film visualisation of test-section top and side walls (flow is from left to right). $\text{S}$ : saddle,  $\text{S}^{\prime }$ : half saddle, $\text{N}$ : node,  $\text{N}^{\prime }$ : half node, $\text{F}$ : focus.

On the top surface, the boundary-layer separation is characterised by a wide zone of very low shear stress where the oil is stagnating and the flow details are difficult to interpret. The directions of the limiting streamlines upstream and downstream of this zone impose the presence of a saddle point ( $\text{S}_{2}$ ) near the centreline. Close to the side walls, two slowly rotating vortices ( $\text{F}_{1}$ and $\text{F}_{2}$ ) can be identified by watching the oil move on the surface. These vortices are reminiscent of the ‘corner vortices’ observed by de Brederode & Bradshaw (Reference de Brederode and Bradshaw1972) in their backward-facing step flow and by Ruderich & Fernholz (Reference Ruderich and Fernholz1986) in a fence and splitter-plate flow. The detachment line is terminated on the side walls by saddle points $\text{S}_{1}$ and $\text{S}_{1}^{\prime }$ on each side of the test section. These saddle points are located in the corners of the test section and separate the wall streamlines on the top surface from those on the side walls. The two streamlines originating from $\text{S}_{1}$ connect to the foci $\text{F}_{2}$ and $\text{F}_{3}$ . The latter focus represents the ‘centre’ of the recirculation region on the side wall.

In contrast to the detachment line, the reattachment line on the top surface can be observed relatively clearly owing to the larger mean shear stress. This line is nearly straight and symmetric with saddle point $\text{S}_{3}$ separating the left and right sides of the test surface. The reattachment line connects to nodes $\text{N}_{1}$ and $\text{N}_{1}^{\prime }$ in the corners of the test section. Interestingly, the termination of the reattachment line on the side wall is not node $\text{N}_{1}$ but saddle point $\text{S}_{4}$ , which is clearly located on the side wall below the corner. This combination of a node in the corner and a saddle on the side wall is consistent with the visualisations of de Brederode & Bradshaw (Reference de Brederode and Bradshaw1972) in the reattachment zone of a backward-facing step. Ruderich & Fernholz (Reference Ruderich and Fernholz1986) also observed this pattern in their fence and splitter-plate flow but interpreted the directions of the limiting streamlines as the juxtaposition of a half-node and a half-saddle, probably because of the small distance separating the two critical points.

The effect of the boundary-layer bleed can be clearly seen on the side wall streamlines. Saddle point $\text{S}_{2}^{\prime }$ separates the streamlines leaving the test section out of the bleed from those continuing towards the exit of the wind tunnel. Nearly all streamlines originating from the upstream part of the test section are swallowed by the bleed. This rather dramatic effect is due to the low momentum of the fluid particles close to the wall. A few millimetres away from the side wall, only a small proportion of the incoming flow near the bottom surface is in fact swallowed by the bleed.

The critical points presented in figure 2 are consistent with the topological constraints of a smooth vector field. Using the terminology of Foss (Reference Foss2004), who builds on the work of Hunt et al. (Reference Hunt, Abell, Peterka and Woo1978), the combination of the top surface and the side wall in figure 2 is the topological equivalent of a collapsed sphere with three holes. The holes in the sphere are the upstream flow, the bleed and the downstream flow. Accordingly, the Euler characteristic ${\it\chi}$ of the collapsed sphere is:

The Poincaré–Hopf theorem requires that the Euler characteristic of a surface be equal to the sum of the indices of all singular points on this surface, bearing in mind that in the case of a collapsed sphere, the points that are located within the interior must be counted twice. In figure 2 there are one node, three foci and four saddles in the interior domain, as well as one half-node and two half-saddles on the surface seams. Therefore we have:

and the topological constraints are satisfied.

The pattern of wall streamlines on the top surface is also consistent with the exact theory of steady, three-dimensional flow separation by Surana, Grunberg & Haller (Reference Surana, Grunberg and Haller2006). Using their terminology, the separation line is a saddle–spiral connection (type S1), while the reattachment line is a saddle–node connection (type R2). These two types of separation/attachment lines are the most commonly observed of only four possible separation and attachment patterns (Surana et al. Reference Surana, Grunberg and Haller2006).

We now turn our attention to the characteristics of the incoming turbulent boundary layer and the turbulence level in the wind tunnel. The boundary layer at $x=1.10$  m was investigated by Mohammed-Taifour et al. (Reference Mohammed-Taifour, Schwaab, Pioton and Weiss2015a ,Reference Mohammed-Taifour, Weiss, Sadeghi, Vétel, Jondeau and Comte-Bellot b ) with a hot-wire anemometer (HWA). On the test-section centreline, the boundary-layer thickness is ${\it\delta}_{99}=28$  mm, the momentum thickness is ${\it\theta}=3.0$  mm, the friction coefficient (obtained from Clauser’s method) is $c_{f}=3.1\times 10^{-3}$ and the friction velocity is $U_{{\it\tau}}=0.98~\text{m}~\text{s}^{-1}$ . This results in a Reynolds number, based on momentum thickness, of $Re_{{\it\theta}}\simeq 5000$ for a reference velocity of $U_{ref}=25~\text{m}~\text{s}^{-1}$ . Across the span of the test section, the integral thicknesses are constant within 5 % and the friction coefficient within 7 %. Figure 3 shows a logarithmic wall-law plot of $u^{+}=\overline{U}/U_{{\it\tau}}$ versus $y^{+}=yU_{{\it\tau}}/{\it\nu}$ , where ${\it\nu}$ is the kinematic viscosity of air. These data were obtained at $x=1.27$  m with a hot-wire anemometer as well as with the PIV system that will be described in § 2.2. There is approximately a decade of logarithmic region, which is typical for a ZPG turbulent boundary layer at this Reynolds number (De Graaff & Eaton Reference De Graaff and Eaton2000).

Figure 3.

Logarithmic wall-law plot of the incoming boundary layer, $x=1.27$  m. Von Kármán constant ${\it\kappa}=0.41$ .

The power spectral density (PSD) of the velocity fluctuations in the potential flow of the wind tunnel ( $x\simeq 0.5$  m, $y\simeq 0.08$  m) is presented in figure 4. These data were measured with an X-wire probe. Note that the large peak that can be observed near 400 Hz does not depend on the velocity and is believed to be caused by electric perturbations in the signal. The vertical ( $v^{\prime }$ ) and lateral ( $w^{\prime }$ ) components show a relatively flat frequency distribution until their roll-off above 200 Hz. On the other hand, the longitudinal ( $u^{\prime }$ ) component shows a much higher energy level for frequencies below 30 Hz. This low-frequency component of $u^{\prime }$ is relatively common in low-speed wind tunnels and is caused by coherent mass-flow fluctuations originating in the centrifugal blower. As described in detail in Weiss et al. (Reference Weiss, Mohammed-Taifour and Schwaab2015), the mass-flow fluctuations impose the use of a filtering technique for accurate measurements of wall-pressure fluctuations. Integrating the PSDs of figure 4 leads to relative turbulence intensities of $\sqrt{\overline{u^{\prime ^{2}}}}/U_{ref}\simeq 0.30\,\%$ , $\sqrt{\overline{v^{\prime ^{2}}}}/U_{ref}\simeq 0.07\,\%$ , and $\sqrt{\overline{w^{\prime ^{2}}}}/U_{ref}\simeq 0.07\,\%$ in the test section for $U_{ref}=25~\text{m}~\text{s}^{-1}$ .

Figure 4.

PSD of velocity fluctuations in the potential flow region ( $x\simeq 0.5$  m,  $y\simeq 0.08$  m,  $z=0$  m).

2.2 High-speed PIV system

Measurements of the longitudinal $U(x,y,t)$ and vertical $V(x,y,t)$ components of the velocity field in the centreline plane ( $z=0$ ) of the TSB were performed using a high-speed PIV system. Figure 5 shows a sketch of the wind tunnel with the PIV arrangement. The laser sheet of approximately 1 mm thickness was oriented vertically and illuminated the centreline plane of the test section. It was created by a CW-diode pumped, Q-switched, double Nd:YLF high-repetition-rate laser (Litron LDY304) generating two pulses of 30 mJ each at a wavelength ${\it\lambda}=527$  nm. Images were recorded using two high-speed CMOS cameras (Phantom V9.1, 12 bits, $11.5~{\rm\mu}\text{m}$ pixel pitch) synchronised with the laser pulses by the use of a LaVision High Speed Controller. During the measurements, the complete wind tunnel circuit was seeded with micro-sized droplets of Bis (2-ethylhexyl) sebacate oil of $1~{\rm\mu}\text{m}$ average size generated by a LaVision seeding system.

Figure 5.

Experimental set-up with the high-speed PIV arrangement. Distances in metres.

The two cameras were placed side by side at a distance of approximately 400 mm from the centreline plane. They were calibrated simultaneously with the help of a LaVision 204-15 calibration plate and were operated with a resolution of 1200 $\times$ 800 pixels. Two 50 mm, $f^{\#}$ 2 Micro Nikkor lens were used for imaging, which provided a field of view of 110 mm $\times$ 74 mm for each camera and an image magnification factor $M\sim 0.125$ . An overlap of 10 mm was maintained in order to create a single vector field by stitching the two images. This led to a total field of view of 200 mm $\times$ 74 mm. The pair of cameras was placed sequentially at six different streamwise positions (slices), as shown in figure 1, in order to cover the complete TSB. At each slice, five separate sequences of 3580 double-frame images were acquired at a frame rate of 900 Hz by each camera. This resulted in five time-resolved sequences of approximately 4 s each in the stitched field of view. The total number of image pairs at each slice was thus 17 900. It was verified that the frame rate of 900 Hz is high enough to capture the low- and medium-frequency unsteadiness in the TSB without aliasing. It is too low, however, to accurately measure small-scale, high-frequency turbulent fluctuations in the flow.

The images were processed by the LaVision DaVis software (version 8.2) using a multi-pass correlation technique with 50 % overlap for enhanced spatial resolution. The initial interrogation window of $48\times 48$ pixels was reduced to $24\times 24$ pixels in a second pass. This resulted in a vector spacing of 1.1 mm in the object plane. Bad vectors with a signal-to-noise ratio below 1.2 were removed and replaced by the median filter. The maximum image-plane displacement corresponding to the free stream velocity $U_{ref}$ was approximately 5 pixels. Observation of the displacement histograms confirmed the absence of peak-locking effect. Based on the analysis of Adrian & Westerweel (Reference Adrian and Westerweel2010), the uncertainty in the measured displacement is approximately 0.1 pixel unit, which translates into a relative uncertainty of 2 % in the measured free stream velocity.

4.1 Breathing motion

Figure 13 shows the contribution of the first 50 POD modes to the TKE in the fourth PIV slice. This specific slice contains the average reattachment position (see figure 1) but a similar distribution was obtained in all other slices. Figure 13 shows that the first 50 modes capture 80 % of the total TKE. Quite interestingly, the first POD mode taken by itself contributes to 31 % of the TKE, whereas the individual contribution from higher modes is much lower and decreases rapidly. A similar behaviour was already observed by Thacker et al. (Reference Thacker, Aubrun, Leroy and Devinant2013) in an incompressible, geometry-induced TSB and by Humble et al. (Reference Humble, Scarano and Van Oudheusden2009) in a shock-induced TSB. In these two cases the contribution of the first POD mode to the total TKE was 28 % and 20 %, respectively. Both groups of researchers have thus suggested that the first POD mode is related to a physical mechanism that is different from the other modes. In particular, Thacker et al. (Reference Thacker, Aubrun, Leroy and Devinant2013) convincingly demonstrated that the first POD mode is exclusively associated with the flapping of their geometry-induced TSB. Similarly Humble et al. (Reference Humble, Scarano and Van Oudheusden2009) associated the first POD mode with the low-frequency unsteadiness in their shock-induced TSB.

Figure 13.

Contribution of the first 50 POD modes to the TKE in the fourth PIV slice.

The first POD mode ${\it\Phi}_{u}^{1}(\boldsymbol{x})$ of the longitudinal velocity is presented in figure 14. The mode was computed sequentially over the six PIV slices and normalised so that the same colour coding could be used in a single field of view. There are obviously some discontinuities between the different slices. Nevertheless, the spatial mode clearly encompasses the complete separation bubble, which indicates that the associated velocity fluctuations are the result of a ‘global’ motion of the TSB. This is consistent with the results obtained by Humble et al. (Reference Humble, Scarano and Van Oudheusden2009) and Thacker et al. (Reference Thacker, Aubrun, Leroy and Devinant2013).

Figure 14.

Normalised first POD mode ${\it\Phi}_{u}^{1}$ of longitudinal velocity on PIV slices 1–6. Grey windows indicate the borders of the PIV slices.

To further investigate the behaviour of the first POD mode in the context of the present experiment, a low-order model of the fluctuating velocity field is reconstructed using only the first POD mode. That is, for each PIV slice, we build the reconstructed field:

Because the complete TSB spans a total of six PIV slices, the PIV data could not be taken simultaneously at each slice and were therefore taken sequentially. This means that the time coefficients $a^{1}(t)$ from each slice are not synchronised and cannot be used to build one time-resolved fluctuating field that completely spans the TSB. However, it is possible to visualise the complete TSB at specific instants defined by a constant value of the time coefficient. For example, we can choose to visualise the reconstructed field by choosing the minimum or maximum of $a^{1}(t)$ at each slice. This technique was already used by Humble et al. (Reference Humble, Scarano and Van Oudheusden2009) and Thacker et al. (Reference Thacker, Aubrun, Leroy and Devinant2013) to visualize extreme flow topologies in their respective separated flows.

Figure 15.

Contour map of $\widetilde{U}/U_{\infty }$ . Flow topologies associated with the first POD mode time-coefficients (a) $(a^{1})_{min}$ , (b) $a^{1}\simeq 0$ , and (c) $(a^{1})_{max}$ . Solid white line shows $\widetilde{U}=0$ .

Figure 15 shows the longitudinal velocity field $\widetilde{U}(\boldsymbol{x},t)$ reconstructed using only the first POD mode for three specific values of the time coefficient: $(a^{1})_{min}$ , $a^{1}\simeq 0$ , and $(a^{1})_{max}$ . The figure is an amalgamation of the 6 PIV slices and some discontinuities can again be observed between the slices. Therefore, the $\widetilde{U}=0$ line was smoothed for ease of visualisation. Notwithstanding these small imperfections, the figure shows a spectacular variation in the size of the TSB. Comparing figure 15(b) with figure 7(a) clearly shows that when $a^{1}(t)\simeq 0$ , the reconstructed field is a good approximation of the average TSB, which is expected from (4.7). The two other fields, figure 15(a,c), can be viewed as conditional averages on the instants when the TSB is highly contracted (when $a^{1}(t)=(a^{1})_{min}$ ) and on the instants when the TSB is largely expanded (when $a^{1}(t)=(a^{1})_{max}$ ). A value of $a^{1}(t)$ between these two extremes would correspond to a TSB with a size comprised between those depicted in figure 15(a,c). Thus, the TSB indeed appears to be ‘breathing’, as suggested by the thermal tuft measurements of Weiss et al. (Reference Weiss, Mohammed-Taifour and Schwaab2015). Quite interestingly, the contour plots presented in figure 15 resemble those obtained by Piponniau et al. (Reference Piponniau, Dussauge, Debiève and Dupont2009), who documented the breathing of their shock-induced TSB using conditional averages of the velocity fields. This suggests that strong similarities exist between low-speed pressure-induced TSBs and high-speed shock-induced TSBs, as already argued by Weiss et al. (Reference Weiss, Mohammed-Taifour and Schwaab2015).

The range of variation in TSB size is surprisingly large: the detachment appears to move between $x^{\ast }=-0.30$ (the ID point) and $x^{\ast }=0.25$ while the reattachment moves between $x^{\ast }=0.85$ and $x^{\ast }=1.20$ (the CR point). The total variation in TSB length caused by the breathing motion is therefore equivalent to 90 % of its average length. It should be kept in mind, though, that the reconstructed fields of figure 15 are never observed in reality. They represent the unsteady skeleton of the TSB over which the turbulence has been averaged out by neglecting higher-order modes. Physically, this can be reconciled by assuming that the large-scale breathing motion observed in figure 15 is of much lower frequency than the turbulent motions in the flow. In other words, the first POD mode is not only characterised by its high energy (by construction) but also by its low frequency. This is consistent with the wall-pressure PSDs presented in figures 11 and 12, where it can be seen that the maximum energy of the pressure signature is indeed present at very low frequencies.

To investigate the time scales associated with the breathing motion, synchronised PIV and wall-pressure measurements were performed. For these experiments, the PIV field of view ranged from $x^{\ast }=-0.43$ to $x^{\ast }=0.04$ (i.e. the cameras were positioned between slices 1 and 2) and the piezo-resistive pressure transducer was positioned at $x^{\ast }=-0.37$ . The latter position is very near the local maximum of $c_{p^{\prime }}$ (see figure 10) as well as the ID point (see figure 6) . The signal of the pressure transducer was corrected based on the signal of a reference transducer placed upstream in the test section using the optimal filtering technique of Naguib, Gravante & Wark (Reference Naguib, Gravante and Wark1996). This removed the spurious pressure fluctuations caused by the wind tunnel blower (see Weiss et al. (Reference Weiss, Mohammed-Taifour and Schwaab2015) for details). To synchronise the pressure transducers and the PIV acquisition, a 5 V TTL (transistor–transistor logic) rising edge pulse was used. At first, both the transducers and the PIV system were armed to start acquiring once they received an external trigger signal. Then, the digital signal was sent out, which triggered the pressure transducer acquisition immediately and the PIV after one second of delay. The PIV images were acquired at a frame rate of 900 Hz for 4 s and the pressure signals at a sampling frequency of 12.8 kHz.

Figure 16.

Normalised time traces of pressure fluctuations at $x^{\ast }=-0.37$ and modal coefficient $a^{1}(t)$ .

Figure 17.

Magnitude-squared coherence function between the wall-pressure fluctuations measured at $x_{1}^{\ast }=-0.37$ and $x_{2}^{\ast }=1.50$ .

Figure 16 shows the normalised time traces of the pressure fluctuations and the time coefficient $a^{1}(t)$ of the first POD mode for the 4 seconds of measuring time. The high-frequency fluctuations of both time traces were removed by a moving-average filter in order to concentrate on the low-frequency character of the signals. Clearly, the signals are highly correlated, with a correlation coefficient of approximately $+0.8$ . This shows that the low-frequency pressure signature observed in figures 11 and 12 is indeed related to the breathing of the TSB. The peaks of the $a^{1}(t)$ signal correspond to an expanded bubble and the valleys to a contracted bubble. The pressure near the ID point thus increases when the bubble expands and it decreases when the bubble contracts.

Figure 17 shows the coherence function between the wall-pressure fluctuations measured at $x_{1}^{\ast }=-0.37$ (near ID) and $x_{2}^{\ast }=1.50$ (just downstream of CR), that is, the signals that correspond to the two main low-frequency zones in figure 12. The coherence is very high in the low-frequency range, which suggests that the two signals are both signatures of the same phenomenon. Low-pass filtering and correlating the time traces also leads to a correlation coefficient of $+0.8$ , which indicates that when the pressure increases near ID, it also increases near CR, and vice versa. Thus, an expanded bubble increases the wall pressure on both sides of the TSB, while a contracted bubble decreases it. This behaviour is consistent with the shape of the static-pressure distribution in the test section. Indeed, in figure 18, we present the sketch of a hypothetical variation in pressure coefficient $c_{p}$ that could occur when the TSB expands and contracts. In this figure, only the curve labelled ‘average bubble’ was actually measured. The ‘small bubble’ and ‘large bubble’ curves are hypothetical departures from the average $c_{p}$ . According to figure 18, an expansion of the TSB must result in a widening of the $c_{p}$ curve, which implies a strong increase in pressure near ID and just downstream of CR. Similarly, a contraction of the TSB implies a significant decrease of the pressure at these positions. This simple model explains why the pressure signature of the breathing motion is mostly observed upstream and downstream of the average TSB.

Figure 18.

Sketch of pressure distribution for different sizes of TSB. Only the ‘average bubble’ curve was actually measured. Other curves are hypothetical departures from the average $c_{p}$ .

4.2 Shedding motion

Beside the low-frequency breathing described above, the TSB is also characterised by a medium-frequency convective motion that is responsible for the global maximum of wall-pressure fluctuations in figure 10. Its dominant Strouhal number is $St_{2}\simeq 0.35$ . Because of its convective nature, and by analogy with fixed-separation flows, Weiss et al. (Reference Weiss, Mohammed-Taifour and Schwaab2015) concluded that this unsteadiness is caused by large-scale vortices traveling in the shear layer and shed downstream of the TSB. Figure 19 shows the PSD of the average out-of-plane vorticity computed on PIV slices 1–4 over approximately 20 s of signal. The PSD amplitude near $St_{2}=0.35$ is growing with the downstream distance, which is consistent with the wall-pressure PSDs shown in figure 11 and the PSDs of streamwise velocity measured by Weiss et al. (Reference Weiss, Mohammed-Taifour and Schwaab2015). This suggests that the unsteadiness is indeed linked to vortical structures in the flow.

Figure 19.

PSD of average out-of-plane vorticity in slices 1 to 4.

Figure 20.

Contours of cross-correlation coefficient $R_{u^{\prime }u^{\prime }}({\rm\Delta}x,{\it\tau})$ for streamwise velocity fluctuations calculated on the maximum loci of ${\it\omega}_{z}$ . Dashed line is a linear fit. Contour levels are from 0.1 to 0.9 with increment of 0.05.

Figure 21.

Normalised convection velocity along the TSB.

Figure 20 shows the cross-correlation coefficient $R_{u^{\prime }u^{\prime }}({\rm\Delta}x,{\it\tau})$ calculated with the streamwise velocity fluctuations at the maximum loci of ${\it\omega}_{z}^{\ast }$ for each PIV slice. The reference signal is located at the midlength of each field of view. ${\rm\Delta}x$ is the longitudinal separation between the two measurement points and ${\it\tau}$ is the time delay between the two signals. The data were averaged over approximately 20 s of data. The isocontours of the cross-correlation coefficient have an elliptic shape with a positive slope that is equal to the inverse of the average convection velocity $U_{c}$ in each PIV slice. $U_{c}/U_{ref}$ is plotted in figure 21 as a function of $x^{\ast }$ and compared to the average velocity measured on the lines of maximum ${\it\omega}_{z}^{\ast }$ and of maximum $\sqrt{\overline{u^{\prime ^{2}}}}/U_{\infty }$ . The good agreement between each plot indicates that the convection velocity of the dominant energetic structures agrees well with the average velocity at the centre of the shear layer. This was already observed by Hudy, Naguib & Humphreys (Reference Hudy, Naguib and Humphreys2007) in a BFS flow. Within the TSB, $U_{c}$ is quasi-constant with a value of $U_{c}/U_{ref}\simeq 0.3$ , which is the same as the value obtained by Weiss et al. (Reference Weiss, Mohammed-Taifour and Schwaab2015) using wall-pressure cross correlations. This is also consistent with the value of $U_{c}/U_{ref}=0.33$ observed in the DNS of Na & Moin (Reference Na and Moin1998b ).

We now attempt to visualise the large-scale structures responsible for the energetic wall-pressure fluctuations. This is not a trivial task: as the flow is fully turbulent, instantaneous visualisations of the vorticity contours (not shown here), even coupled with local vortex-identification schemes, reveal a variety of structures with a wide range of space and time scales. In order to visualise the large-scale structures only, is necessary to filter out the small-scale turbulent motions. We use two separate techniques to achieve this goal: in the first technique, the local $Q$ criterion introduced by Hunt, Wray & Moin (Reference Hunt, Wray and Moin1988) is applied on POD-filtered data, while in the second one the non-local ${\it\Gamma}_{2}$ function of Graftieaux, Michard & Grosjean (Reference Graftieaux, Michard and Grosjean2001) is used on the original, non-filtered velocity fields. As will be demonstrated below, the two methods give similar results and succeed in identifying the large-scale structures in the shear layer.

We start with the first technique by constructing a low-order model of the velocity field using only POD modes 2–50. As shown in figure 13, this keeps the structures responsible for 80 % of the TKE while neglecting the effect of the breathing motion characterised by the first POD mode. In other words, we concentrate on the medium-frequency fluctuations in the flow. Thus, we consider the reconstructed field $\widetilde{\boldsymbol{U}}(\boldsymbol{x},t)=(\widetilde{U}(x,y,t),\widetilde{V}(x,y,t))$ defined by:

Figure 22 shows contour plots of the second invariant $Q$ of the velocity-gradient tensor $\boldsymbol{{\rm\nabla}}\widetilde{\boldsymbol{U}}$ , together with the convective velocity vector field $(\widetilde{U}-U_{c},\widetilde{V})$ , where $U_{c}=0.4U_{ref}$ . These snapshots were taken in PIV slice 4 and are separated by approximately 2.2 ms. The red patches in the figure correspond to regions of maximum $Q$ , which are typical signatures of vortex cores (Chakraborty, Balachandar & Adrian Reference Chakraborty, Balachandar and Adrian2005). This is confirmed by the shape of the convective velocity field, which clearly indicates a rotational motion in the regions of maximum $Q$ . The snapshots show that a well-defined vortex enters the field of view at $t\simeq t_{0}$ and is convected downstream as $t$ increases. It takes approximately 20 ms to cross the complete field of view, which gives a convection velocity of $U_{c}/U_{ref}=0.4$ that is almost equal to the average convection velocity near $x^{\ast }=1$ plotted in figure 21. A second, less distinct structure appears to be trailing the first one but is not clearly observed in each snapshot. This might be caused by local interactions with smaller structures that alter the rotational character of the velocity field.

Figure 22.

Illustration of vortical flow dynamics in PIV slice 4 with contour map of $Q>0.1$ and convective velocity field $(\widetilde{U}-U_{c},\widetilde{V})$ . Vector field reconstructed from POD modes 2 to 50.

Our second technique consists in computing the ${\it\Gamma}_{2}$ function defined by Graftieaux et al. (Reference Graftieaux, Michard and Grosjean2001):

where $S$ is a square domain centred on point $O$ and where $N$ is the number of velocity measurement points $P\in S$ over which the sum is performed. The domain $S$ plays the role of a spatial filter and its height was chosen equal to the average height of the shear layer. $\overline{\boldsymbol{U}}_{O}$ is the local convection velocity averaged over the domain $S$ and $\boldsymbol{z}$ is the out-of-plane unit vector. ${\it\Gamma}_{2}$ measures the swirl strength when the condition $|{\it\Gamma}_{2}|>2/{\rm\pi}$ is satisfied. Positive values correspond to a clockwise rotation and negative values to a counter-clockwise rotation. The ${\it\Gamma}_{2}$ function was specifically designed to separate the velocity fluctuations caused by large-scale vortices from those related to small-scale turbulence (Graftieaux et al. Reference Graftieaux, Michard and Grosjean2001).

Figure 23 shows contour plots of ${\it\Gamma}_{2}$ for the same snapshots as those presented in figure 22. In contrast to figure 22, ${\it\Gamma}_{2}$ was computed on the original (i.e. non-POD-filtered) velocity fields. Nevertheless, the ${\it\Gamma}_{2}$ function clearly identifies the same dominant counter-clockwise rotating structure that moves through the field of view. The second trailing structure is less clearly observed, though it is obvious in the last snapshot. A third structure is also detected closer to the wall for $t\geqslant t_{0}+6.6~\text{ms}$ , though it is unclear if this is an independent coherent structure or part of the main vortex that moves through the field. Generally speaking, both the $Q$ and ${\it\Gamma}_{2}$ functions succeed in identifying large-scale convective structures in the flow, though we emphasize here that ${\it\Gamma}_{2}$ does so on the original velocity fields whereas $Q$ only provides usable results on the POD-filtered fields.

Figure 23.

Illustration of vortical flow dynamics in PIV slice 4 with contour map of ${\it\Gamma}_{2}\leqslant -2/{\rm\pi}$ and convective velocity field $(U-U_{c},V)$ . Positions where ${\it\Gamma}_{2}>-2/{\rm\pi}$ are set to white.

The ${\it\Gamma}_{2}$ function can also be used to compute the average passage frequency of large-scale structures. To do this we build, at each streamwise position, an indicator function $I(t)$ whose values are 0 when ${\it\Gamma}_{2}>-2/{\rm\pi}$ and 1 when ${\it\Gamma}_{2}\leqslant -2/{\rm\pi}$ . This indicator function has the form of an aperiodic square wave where $I(t)=1$ whenever a large-scale structure travels through the position of interest and $I(t)=0$ otherwise. The number of structures is then obtained by counting the number of positive steps in $I(t)$ and the average passage frequency is computed by dividing the number of structures by the total measurement time. This passage frequency, converted to a Strouhal number $St=fL_{b}/U_{ref}$ , is presented in figure 24 and compared to the dominant Strouhal number obtained from the PSD of wall-pressure fluctuations reported in Weiss et al. (Reference Weiss, Mohammed-Taifour and Schwaab2015). The good agreement between the Strouhal numbers obtained from both methods confirms that the medium-frequency pressure signature observed at $St_{2}\simeq 0.35$ is indeed related to the convection and shedding of large-scale vortices embedded in the shear layer.

Figure 24.

Strouhal number of large-scale coherent structures obtained by three independent methods (see text).

Figure 25.

Normalised vorticity thickness as a function of streamwise position.