2.1 Flow set-up

A water flume with a rectangular cross-section has been modified to form a water tunnel by covering its free surface as shown in the schematic of figure 1(a). The acrylic plate on the top surface of the flume applies a no-slip boundary condition and prevents formation of surface waves. The flow passes through several honeycombs and screens before entering the channel to ensure a uniform flow. A 2D channel with variable cross-section is formed between the vertical sidewall of a water flume and a multi-segment wall with leading-edge (A–B), straight (B–C) and expanding (C–E) sections, as shown in figure 1(a) and the top view of figure 1(b). The dimensions of the set-up are indicated based on the expansion height of the diffuser, $H=165$  mm. The distance $H$ is defined as the distance between points C and E, perpendicular to the flow direction, as seen in figure 1(b). The leading-edge section of A–B is an arc with a $3.63H$ (0.6 m) radius followed by a $15.15H$ (2.5 m) long straight section (B–C) with a small diverging angle of $3.4^{\circ }$ to develop a TBL at zero pressure gradient. The flow stays attached over this curve and the following flat plat as confirmed by tuft visualization. The boundary layer transition to turbulence is forced $0.90H$ (150 mm) downstream of the start of the straight section (point B) using a 1.5 mm diameter wire. The distance between the side wall of the flume and the mid-point of section B–C is $1.52H$ (0.25 m). The depth of the channel ( $W$ ) in the out-of-plane direction of figure 1 is about $3.03H$ (0.5 m). The C–D section is an arc with $3.64H$ (0.6 m) radius connecting to the straight section of the diffuser (D–E), which is a flat plate with $3.03H$ (0.5 m) length and $18.4^{\circ }$ angle with respect to the channel wall. The pressure-induced separation is formed $2H$ (0.33 m) downstream of the start of the diffuser section as indicted in the magnified view of figure 1(c).

The $x$ – $y$ coordinate system is defined with respect to the diffuser plate as indicated in figure 1(b,c). The $x$ -axis is aligned with the D–E section. The origin of the $x$ – $y$ coordinate system is placed at the point where $\unicode[STIX]{x2202}\langle U\rangle /\unicode[STIX]{x2202}y=0$ at the wall. This origin merely serves as a reference for the coordinate system. The flow is intermittent and the instantaneous location of separation moves along the diffuser plate. The expansion height of the diffuser, $H$ , is used to normalize the coordinates as $X=x/H$ , $Y=y/H$ . Another straight section (E–F) is added downstream of the diffuser to prevent interaction of the separated shear layer with the wake. During set-up, tuft visualizations are applied to ensure an attached flow on the expanding arc (section C–D) and also on the wall of the water flume.

The Re number based on the length of section B–C and the centreline velocity is about $8.1\times 10^{5}$ . A relatively thick TBL with a $\unicode[STIX]{x1D6FF}$ thickness of approximately 52 mm forms just before the start of the diffuser curve. The $\unicode[STIX]{x1D6FF}$ is estimated based on Schlichting (Reference Schlichting1979) following

(2.1) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}=0.37lRe_{l}^{-1/5}, & & \displaystyle\end{eqnarray}$$

where $l$ is the distance from the tripping wire to point C, and $Re_{l}$ is based on the velocity at the centre of the channel ( $U_{ref}$ ) and $l$ . The ratio of the depth of the channel to the boundary layer thickness ( $W/\unicode[STIX]{x1D6FF}$ ) is 9.6. However, the thickness of TBL rapidly increases in the expanding section due to the APG. Pitot tube measurements are taken to characterize the velocity and pressure along the centreline of the channel as indicated by $x^{\prime }$ axis in figure 1(b). The origin of $x^{\prime }$ axis is aligned in the streamwise location with $x=0$ as seen in figure 1(c). The velocity upstream of the diffuser at the centreline of the channel is $0.45~\text{m}~\text{s}^{-1}$ , and gradually decreases in the diffuser as shown in figure 1(d). The pressure measurement in figure 1(d) also shows a relatively constant pressure coefficient at $x^{\prime }<3H$ followed by an increase of the pressure coefficient in the diffuser section. The pressure coefficient is defined as $C_{p}=(p-p_{c})/(0.5\times \unicode[STIX]{x1D70C}\times U_{ref}^{2})$ , where $p_{c}$ and $U_{ref}=0.45~\text{m}~\text{s}^{-1}$ are the static pressure and velocity at $x^{\prime }=-10H$ , respectively. The inner scaling of the TBL at point C (just before the start of the diffuser curve) can be obtained from an estimation of the wall shear stress following

(2.2) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70F}=0.0296\unicode[STIX]{x1D70C}U_{ref}^{2}Re_{l}^{-1/5}, & & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D70C}$ is density of water (Schlichting Reference Schlichting1979). The estimated friction velocity and wall unit are $u_{\unicode[STIX]{x1D70F}}=0.0193~\text{m}~\text{s}^{-1}$ and $\unicode[STIX]{x1D706}=52~\unicode[STIX]{x03BC}\text{m}$ , respectively.

Figure 2.

A schematic view of the planar and tomo-PIV set-ups. (a) The planar PIV set-up consists of four cameras in staggered configuration imaging the FOV through front surface mirrors. (b) The tomo-PIV set-up includes a combination of cylindrical and spherical optics to form a collimated laser sheet. The four cameras are arranged in a cross configuration. Cameras 2 and 4 have the largest solid angle ( ${\sim}$ 35) and image through a water-filled prism mounted on the wall of the channel (not shown here for clarity).

2.2 Planar PIV

The separated flow is characterized using four PIV cameras, which simultaneously capture the velocity field upstream and downstream of the detachment locations in the mid-span of the plate. The combined FOV of the four CCD cameras covers ${\sim}1.45H$ (240 mm) in the streamwise direction ( $x$ ) and ${\sim}0.4H$ (66 mm) in the wall-normal direction ( $y$ ) as shown in figure 2(a). Each PIV camera (Imager ProX, LaVision GmbH) has a $2048\times 2048$  pixels sensor with a pixel size of $7.4~\unicode[STIX]{x03BC}\text{m}\times 7.4~\unicode[STIX]{x03BC}\text{m}$ and a 14-bit resolution. The FOV of each camera is ${\sim}75~\text{mm}\times 75~\text{mm}$ with a digital resolution of about $27~\text{pixels}~\text{mm}^{-1}$ , which varies slightly from one camera to another. Each camera is equipped with an SLR lens with a focal length of $f=105$  mm at an aperture setting of $f/11$ . The estimated depth of field is ${\sim}$ 12 mm. The four PIV cameras are arranged in a staggered configuration and image through four front surface mirrors oriented at $45^{\circ }$ angle with respect to the line of sight of the cameras as shown in figure 2(a). This configuration allows the FOV of the side-by-side cameras at a magnification of 0.2 to overlap and form a continuous FOV. A calibration process using a 2D target is applied to correct for any optical and perspective distortion of the images, and later stitch the vector fields.

The FOV is illuminated by a Nd:YAG laser (Spectra-Physics, PIV400) with a maximum output power of $400~\text{mJ}~\text{pulse}^{-1}$ at 532 nm wavelength. A combination of spherical and cylindrical lenses generate a laser sheet with ${\sim}$ 1 mm thickness and 300 mm width. The laser sheet and FOV are located $1.5H$ above the bottom wall of the flume at the middle of the plate. The flume is seeded with $d_{p}=2~\unicode[STIX]{x03BC}\text{m}$ silver-coated spherical glass particles (SG02S40, Potters Industries) with a relative density of $\unicode[STIX]{x1D70C}_{p}=2.6~\text{g}~\text{cm}^{-3}$ . The response time of the tracers ( $\unicode[STIX]{x1D70F}_{p}=\unicode[STIX]{x1D70C}_{p}d_{p}^{2}/18\unicode[STIX]{x1D707}$ , $\unicode[STIX]{x1D707}$ in the dynamic viscosity of the water) with respect to the inner time scale of the TBL ( $\unicode[STIX]{x1D70F}_{f}$ ) is characterized using the Stokes number (Stk) of the tracers, $Stk=\unicode[STIX]{x1D70F}_{p}/\unicode[STIX]{x1D70F}_{f}$ . The inner time scale of the flow is estimated from the wall-shear stress obtained using (2.2). The estimated Stk is ${\sim}2\times 10^{-4}$ , which indicates that the tracers follow the smallest eddies of the flow (i.e.  $Stk\ll 1$ ). Similar tracers have previously been used for detailed measurement of the inner layer at high-Reynolds turbulent channel flows using digital holographic microscopy by Talapatra & Katz (Reference Talapatra and Katz2012) and Ling et al. (Reference Ling, Srinivasan, Golovin, McKinley, Tuteja and Katz2016). The cameras and the laser system are synchronized using a programmable timing unit (PTU9, LaVision GmbH). The separation time between the two laser pulses is 2 ms and data acquisition is performed at 2 Hz. An ensemble of 7000 double-frame images is captured using each camera to ensure statistical convergence.

To improve the signal-to-noise ratio (SNR) in the near-wall region, the minimum intensity of the ensemble of images is subtracted from each image. The images are then normalized by their ensemble average. The double-frame images are cross-correlated with a commercial software (DaVis 8.3, LaVision GmbH) using a multi-pass algorithm with a final window size of $32\times 32$  pixels ( $1.18\times 1.18~\text{mm}^{2}$ ) and 75 % overlap. This results in a spatial dynamic range of ${\sim}$ 100 in the streamwise direction. The first valid vector from the wall is located at $Y=0.002$ . The universal outlier detection method of Westerweel & Scarano (Reference Westerweel and Scarano2005) is applied to detect the spurious vectors. Less than 2 % of the vectors are detected as outliers and replaced through linear interpolation. A summary of the specifications of the PIV system is provided in table 1. The uncertainty of the PIV system is estimated in Davis 8.4 (LaVision GmbH) using the method of Wieneke (Reference Wieneke2015) from statistical properties of the correlation peak. This method is based on the intensity difference of the first image frame and the second frame dewarped back using the computed displacement field. Inspection of the uncertainty fields shows that maximum local uncertainty within the FOV of all four cameras is ${\sim}0.008~\text{m}~\text{s}^{-1}$ (0.4 pixels), which is about 1.7 % of $U_{ref}$ .

Table 1.

Imaging and processing parameters of the planar PIV and tomo-PIV experiments.

The initial stitching process of the four simultaneously acquired vector fields is carried out using the mapping function obtained from the 2D calibration target. An additional rotational and translational correction is applied using a custom function in MATLAB (MathWorks Inc.) based on cross-correlation of the velocity in the overlapped region of neighbouring cameras. An uncertainty analysis over the ensemble of the vectors shows that the maximum difference between the velocity magnitudes in the overlap region of any two cameras is ${\sim}6\times 10^{-3}~\text{m}~\text{s}^{-1}$ (0.3 pixels). A sample stitched instantaneous vector field from planar PIV is shown in figure 3(a). As is observed, there is no visible boundary between the vector fields of the four cameras. The vector field shows an upstream forward flow up to about $X=-0.3$ , where a large backflow region is identified with the contour enclosed by the solid line. There are also smaller backflow regions at about $X=-0.8$ and $X=+0.1$ .

Figure 3.

(a) An instantaneous flow field captured using the four-camera planar PIV. The contours of the backflow region are enclosed by a solid line. The thick solid contour line shows the boundaries of the backflow region based on $\langle U\rangle =0$ . (b) Profiles of instantaneous velocity ( $U$ ) and the low-pass filtered velocity ( $U_{f}$ ) are shown along $Y=0.025$ . The filtered signal ( $U_{f}$ ) only retains the large backflow region between $X=-0.25$ and $-0.07$ . The $X=-0.25$ and $X=-0.07$ points are separation and attachment locations, respectively.

2.3 Tomo-PIV

The 3D characterization of the flow field in the separation zone is carried out using a tomo-PIV system (Elsinga et al. Reference Elsinga, Scarano, Wieneke and van Oudheusden2006). The laser light from the same Nd:YAG laser as used for the planar PIV was shaped into a collimated laser sheet with 12 mm thickness and 100 mm width using a combination of spherical and cylindrical optics. The laser sheet was directed parallel to the diffuser surface from the bottom of the channel using a large, high-power mirror, as shown in figure 2(b). The laser sheet is carefully directed flush to the surface to minimize wall reflection, covering up to the wall-normal location of $y=12$  mm. Knife-edge filters are installed on the bottom glass wall of the channel to crop the low-energy tails of the sheet and provide a relatively top-hat intensity profile.

The same cameras as used for the planar PIV, equipped with the same $f=105$  mm SLR lens with the addition of Scheimpflug adapters, are arranged in a cross-like orientation as shown in figure 2(b). Cameras 2 and 4 have larger viewing angles and image through water-filled prisms installed on the vertical glass wall of the flume to reduce image distortion and astigmatism of particle images. The aperture of the SLR lens on camera 3 is set at $f/16$ to collect more light from its backward scattering configuration, while the three other cameras image the FOV at an aperture setting of $f/22$ . The depth of field of camera 3 with the smallest aperture of $f/16$ is ${\sim}$ 23 mm, which covers the entire thickness of the laser sheet. The digital resolution of the 3D imaging system is $27.3~\text{voxels}~\text{mm}^{-1}$ with a measurement volume of $(\unicode[STIX]{x0394}x,\unicode[STIX]{x0394}y,\unicode[STIX]{x0394}z)=75~\text{mm}\times 12~\text{mm}\times 70~\text{mm}$ , where $z$ is the spanwise direction as shown in figure 2(b). The cameras are synchronized with the laser system using a programmable timing unit (PTU9, LaVision, GmbH). An initial mapping function is obtained using a pinhole model by imaging a 3D calibration target traversed four times in 2 mm steps in the $y$ direction. The volume self-calibration technique is iteratively applied in Davis 8.3 (LaVision GmbH) to reduce the uncertainty of the calibration function to less than 0.1 pixels (Wieneke Reference Wieneke2008).

The water flume is seeded with $2~\unicode[STIX]{x03BC}\text{m}$ silver-coated tracer particles with particle images between 3 and 6 pixels. The particle concentration in the images is about 0.05 particles per pixel (ppp) with a source density ( $N_{s}$ ) of 0.31. The SNR of the images is improved by subtracting the minimum of the ensemble images from each image and normalization of the images by the ensemble average. Subtracting the local minimum intensity and normalizing the signal over a kernel of 50 pixels further improves the images. The volume reconstruction is performed using the MART algorithm (Herman & Lent Reference Herman and Lent1976) in Davis 8.3. Subsequently, multi-pass volumetric cross-correlation with a final interrogation volume of $48\times 48\times 48$ voxels ( $1.59~\text{mm}\times 1.59~\text{mm}\times 1.59~\text{mm}$ ) with 75 % overlap is applied. The calculated vector fields are further processed to remove spurious vectors using the universal outlier detection (Westerweel & Scarano Reference Westerweel and Scarano2005). The first valid data point of the tomo-PIV is located at $Y=0.005$ .

To calculate the uncertainty of tomo-PIV in an incompressible flow, standard deviation ( $\unicode[STIX]{x1D70E}$ ) of the divergence of the velocity vector ( $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{V}$ ) was applied by Scarano & Poelma (Reference Scarano and Poelma2009) over the ensemble of data as $\unicode[STIX]{x1D700}=\langle \unicode[STIX]{x1D70E}(\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{V})\rangle$ . They reported an $\unicode[STIX]{x1D700}$ of 0.01 and $0.04~\text{voxels}~\text{voxel}^{-1}$ for their unfiltered data in the wake of a cylinder at $Re=180$ and 1080, respectively. Kim, Westerweel & Elsinga (Reference Kim, Westerweel and Elsinga2012) performed high-resolution tomo-PIV experiments on a confined liquid droplet and reported $\unicode[STIX]{x1D700}=0.025~\text{voxels}~\text{voxel}^{-1}$ uncertainty. Atkinson et al. (Reference Atkinson, Coudert, Foucaut, Stanislas and Soria2011) also reported $\unicode[STIX]{x1D700}=0.05~\text{voxels}~\text{voxel}^{-1}$ for tomo-PIV measurements in a TBL. The estimated $\unicode[STIX]{x1D700}$ of the current experiment is about $0.01~\text{voxels}~\text{voxel}^{-1}$ , which is in agreement with previous tomo-PIV experiments. A summary of the system parameters of the tomo-PIV is provided in table 1.

2.4 Conditional sampling of flow separation

A conditional sampling algorithm is applied to detect the occurrence of flow separation in instantaneous velocity fields obtained from planar and tomo-PIV. The detection criterion is based on identification of points with zero instantaneous streamwise velocity ( $U=0$ ) when streamwise velocity transitions from an upstream positive $U$ to a downstream negative $U$ at a wall-normal distance of $Y=0.025$ . This criterion indicates that $\text{d}U/\text{d}y$ is approximately zero at the wall. However, it does not ensure zero wall shear stress due to possibility of a spanwise velocity component. The sampling algorithm is further explained using the instantaneous planar PIV sample shown in figure 3(a). The streamwise velocity ( $U$ ) from the planar PIV along the $Y=0.025$ line is shown in figure 3(b). As is observed, the signal contains both small- and large-scale fluctuations. Therefore, the signal is first low-pass filtered using a moving window to avoid detection of small turbulent structures. The applied conditional averaging is intended to only detect the large regions of backflow with a length scale of the same order of magnitude as the thickness of the upcoming boundary layer. The moving window is $0.12H$ (19.8 mm) in the $x$ direction and is shifted along $Y=0.025$ . As a result, the filtered velocity ( $U_{f}$ ) in figure 3(b) only has large-scale fluctuations. The unfiltered signal in figure 3(a) has several regions of negative velocity at $X=-0.77,-0.35,-0.15,+0.08$ and $+0.19$ . However, the filtered signal only retains the largest backflow region at $X=-0.15$ . A moving window with a kernel of $(\unicode[STIX]{x0394}X,\unicode[STIX]{x0394}Y,\unicode[STIX]{x0394}Z)=(0.12H,0.02H,0.003H)$ is also applied to the 3D velocity fields of the tomo-PIV. Similar to planar PIV, points with $U=0$ between an upstream forward and a downstream backward flow are sampled. The procedure does not sample flow transition from backward to forward motion, which indicates an attachment point.

The detected $U=0$ point can be at different locations within the FOV. The centre of the conditionally sampled window is positioned at the detected $U=0$ point to spatially align the flow structures before carrying out the averaging procedure. The coordinate system and the averaged quantities obtained from the conditional sampling are indicated using subscript $c$ . Only $U=0$ points, which are within $\pm 0.1H$ of the origin (i.e.  $-0.1<X<0.1$ ), are selected to maintain a large sampling window. The conditional sampling in tomo-PIV data is carried out along the centreline of the volume at $Z=0$ and $Y=0.03H$ .

3.1 Three-dimensional mean velocity field

The contours of the mean velocity field from planar PIV in the $X$ – $Y$ plane is shown in figure 4(a). The velocity field is normalized using the centreline velocity of the upstream channel ( $U_{ref}$ ). The contours show a large region of forward flow and a small region with backward flow at the downstream end of the FOV. Profiles of a velocity vector in the near-wall region are also shown in the figure 4(a) to illustrate the effect of APG on the boundary layer. The vectors demonstrate deceleration of the flow toward $X=0$ and the formation of a recirculation region at $X>0$ .

Figure 4.

(a) Contour of mean streamwise velocity (i.e.  $\langle U\rangle$ ) obtained from planar PIV, showing deceleration of the flow and formation of backflow at the downstream end of the FOV. The contours enclosed by the solid line indicate negative streamwise velocity. The velocity vectors are down-sampled in the $X$ and $Y$ directions, and are only shown at $Y<0.1$ for clarity. The rectangular box with dashed lines specifies the tomo-PIV domain in the $X$ and $Y$ directions. (b) 3D visualization of the streamlines of the mean velocity field from tomo-PIV showing a spiral motion. The isosurface shows the contour of zero mean streamwise velocity ( $\langle U\rangle =0$ ). (c) The streamlines of the mean velocity field from tomo-PIV are shown along with contours of the magnitude of the mean velocity vector (i.e.  $|\langle \boldsymbol{U}\rangle |$ ) at $0.004U_{ref}$ (blue online/dark grey), $0.022U_{ref}$ (green online/medium grey) and $0.044U_{ref}$ (yellow online/light grey).

The tomo-PIV measurement captures a smaller streamwise ( ${\sim}0.5H$ ) and wall-normal extent ( ${\sim}0.08H$ ) in the $X$ – $Y$ plane relative to the planar PIV, as shown by the dashed rectangle in figure 4(a). However, the tomo-PIV measurement covers ${\sim}0.5H$ in the spanwise direction to visualize the 3D motions. The streamlines in figure 4(b) show a spiral motion around an axis in the $Y$ direction. The isosurface in this figure indicates the plane of zero mean streamwise velocity ( $\langle U\rangle =0$ ), separating regions of forward and backward flows. The forward flow region is observed at the side of the FOV with negative $Z$ , and the backward motion is on the side of the FOV with positive $Z$ . The isosurface is tilted with respect to the streamwise direction: it starts at $(X,Z)=(-0.2,+0.1)$ and diagonally continues until $(X,Z)=(+0.2,-0.1)$ . The intersection of the $\langle U\rangle =0$ isosurface with $Z=0$ plane forms the boundary between the forward and backward flows observed in the mean 2D flow of figure 4(a). The 3D measurement confirms that the presence of zero mean wall-shear stress in a 2D plane of the flow does not necessarily indicate flow separation.

The spiral motion of the mean flow and isosurfaces of the magnitude of the mean velocity vector ( $|\boldsymbol{U}|$ ) are shown in figure 4(c). The tangential velocity of this tornado-like vortex increases with radial distance from its centre as observed from the contours. The high velocity at the periphery indicates the ‘forced’ nature of this vortex as opposed to a free vortex with a high-velocity core. The focus point (centre of the vortex) has a negligible velocity magnitude and zero wall-shear stress.

It is important to note that flow fields in figure 4 are obtained by averaging instantaneous vector fields of an unsteady intermittent flow field. Therefore, the spiral motion of the mean flow in figure 4(b,c) can be an artifact of averaging: a complete spiral pattern may not exist in the instantaneous flow fields and may not play a role in the flow dynamics. In the following sections, analysis of the coherent structures of the instantaneous flow field will be carried out using conditional averaging and POD to identify the dominant flow structures.

To visualize the time fraction of the forward flow, the intermittency factor $\unicode[STIX]{x1D6FE}$ from planar PIV is shown in figure 5. The figure shows the dominant presence of the forward flow in the upstream section of the FOV while the backward flow becomes more frequent at the downstream area. The point with zero wall-shear stress (i.e.  $X=0$ ) is located at about $\unicode[STIX]{x1D6FE}=50\,\%$ . The $\unicode[STIX]{x1D6FE}$ distribution presented by Mohammed-Taifour & Weiss (Reference Mohammed-Taifour and Weiss2016) shows transition from forward flow ( $\unicode[STIX]{x1D6FE}\sim 100\,\%$ ) to the separation point ( $\unicode[STIX]{x1D6FE}\sim 50\,\%$ ) over a shorter distance. This is due to the larger pressure gradient of approximately $\unicode[STIX]{x0394}C_{p}/\unicode[STIX]{x0394}x\sim 1.6$ in their experiment relative to the smaller $\unicode[STIX]{x0394}C_{p}/\unicode[STIX]{x0394}x\sim 0.6$ of the current investigation. The contour of the mean velocity in figure 4(a) and the contour of $\unicode[STIX]{x1D6FE}$ in figure 5 are very similar to the patterns observed by Simpson et al. (Reference Simpson, Strickland and Barr1977) and Mohammed-Taifour & Weiss (Reference Mohammed-Taifour and Weiss2016) in 2D separated flows. In fact, no effect is observed from the strong 3D motion in the planar PIV measurements. This highlights the importance of verification of spanwise homogeneity in investigation of 2D separation, as it has been carefully carried out by oil-film visualization in Mohammed-Taifour & Weiss (Reference Mohammed-Taifour and Weiss2016).

Figure 5.

Distributions of intermittency factor $\unicode[STIX]{x1D6FE}$ defined as the time fraction of the forward flow. The contours show a higher probability of backward flow in the downstream region of the FOV. The white solid contour line shows the boundaries of the backward flow region with $\langle U\rangle =0$ .

3.2 Reynolds stresses

The planar PIV measurements are applied here to investigate the distribution of Reynolds stresses in the $XY$ plane in the vicinity of the origin where $\unicode[STIX]{x2202}\langle U\rangle /\unicode[STIX]{x2202}y=0$ . The analysis of this section using planar PIV does not provide a complete characterization since the spanwise component and a volumetric distribution are not available. However, it is useful to compare the results of the planar PIV of the current 3D separation with the previous literature on 2D separation to evaluate the variation of Reynolds stresses near the mean separation point.

The normal and shear Reynolds stresses gradually increase with an increase of the streamwise distance as seen in figure 6. The upper right corner ( $X\sim 0.4$ and $Y\sim 0.3$ ) of figure 6(a,b) have the largest $\langle u^{2}\rangle$ and $\langle v^{2}\rangle$ fluctuations, respectively. This region also has a large level of $-\langle uv\rangle$ , as seen in figure 6(c), and so $u$ and $v$ fluctuations are correlated. The large correlated $u$ and $v$ fluctuations are associated with the roll-up of vortices from the separated shear layer. With the increase of $X$ , the region with large Reynolds stress expands in the $Y$ direction and its peak location moves further away from the wall.

Figure 6.

Distribution of Reynolds stresses (a) $\langle u^{2}\rangle /U_{ref}^{2}$ , (b) $\langle v^{2}\rangle /U_{ref}^{2}$ and (c) $-\langle uv\rangle /U_{ref}^{2}$ . The distributions show a similar pattern with a high turbulence region above the wall. The dashed line in (a) shows the loci of the peak of $-\langle uv\rangle$ at each streamwise location. The solid black line shows the $\langle U\rangle =0$ contour.

The loci of the $-\langle uv\rangle$ peaks at each streamwise location are shown by a dashed line in figure 6(c). The line originates upstream of the mean separation point, due to the intermittency and movement of the separation line. The slope of the dashed line shows that the separated shear layer gradually moves away from the wall. The results are in agreement with the gradual increase and widening of Reynolds stresses observed in the APG-induced separation by Thompson & Whitelaw (Reference Thompson and Whitelaw1985) over the trailing edge of an airfoil and also by Song & Eaton (Reference Song and Eaton2002, Reference Song and Eaton2004) over an arc-shaped ramp. Scarano & Riethmuller (Reference Scarano and Riethmuller1999) also observed a similar trend in Reynolds stress distribution downstream of a backward-facing step with a fixed separation point. The movement of the separation point on the wall does not allow for the formation of a separation bubble with low Reynolds stress in the ensemble averages of figure 6. A negligible Reynolds stress is present inside the fixed separation bubble observed by Simpson (Reference Simpson1989) and Cuvier et al. (Reference Cuvier, Foucaut, Braud and Stanislas2014).

The comparison of the current results with the measurements of Cuvier et al. (Reference Cuvier, Foucaut, Braud and Stanislas2014) and Mohammed-Taifour & Weiss (Reference Mohammed-Taifour and Weiss2016) demonstrates different distributions of Reynolds stresses near the mean separation, due to the geometry at the separation point and the intensity of the pressure gradient. The local high shear at the geometric edge of the ramp model resulted in an initial high value of $\langle u^{2}\rangle$ in Cuvier et al. (Reference Cuvier, Foucaut, Braud and Stanislas2014), while this is not observed in the APG-induced separation of the current investigation or Mohammed-Taifour & Weiss (Reference Mohammed-Taifour and Weiss2016). The larger pressure gradient in Mohammed-Taifour and Weiss’s (Reference Mohammed-Taifour and Weiss2016) experiment rapidly detached and lifted up the shear layer, causing an immediate large $u$ fluctuation. The roll-up of vortices and intensification of $v$ fluctuation occurred more slowly and further downstream. The smaller pressure gradient of the current experiments allows separation and roll-up of the vortices to occur concurrently. Therefore, $u$ and $v$ fluctuations increase together.

3.3 Transport of turbulence

The direction of transport of turbulent kinetic energy in $XY$ plane can be inferred from the third-order moments of turbulence fluctuations. The transport of streamwise turbulent kinetic energy (i.e.  $\langle u^{2}\rangle$ ) by $u$ and $v$ fluctuations is shown in figure 7 using the distribution of $\langle u^{3}\rangle$ and $\langle u^{2}v\rangle$ . The contour plot of $\langle u^{3}\rangle$ in figure 7(a) shows two inclined regions of positive and negative $\langle u^{3}\rangle$ at the bottom and top of the FOV, respectively. The region with the positive $\langle u^{3}\rangle$ in the vicinity of the wall indicates dominant transport of $u^{2}$ with positive $u$ fluctuations. This area expands in the wall-normal direction with an increase of the streamwise distance. The region with a negative $\langle u^{3}\rangle$ , farther away from the wall, indicates dominant transport of $u^{2}$ with negative $u$ fluctuations. To specify the relevant turbulent motion, the $u$ fluctuations are considered in conjunction with the dominant $v$ fluctuations from figure 7(b). The region with negative $\langle u^{2}v\rangle$ close to the wall indicates turbulence transport by negative $v$ while the upper area with positive $\langle u^{2}v\rangle$ indicates dominant transport away from the wall by positive $v$ fluctuation.

Figure 7.

Distribution of (a) $\langle u^{3}\rangle /U_{ref}^{2}$ and (b) $\langle u^{2}v\rangle /U_{ref}^{2}$ to identify the dominant transport of streamwise turbulent kinetic energy. The dashed lines represent the contours of negative $\langle u^{3}\rangle$ and $\langle u^{2}v\rangle$ . The dominant transport of $u^{2}$ is carried out by sweeps in the near-wall region, while the region away from the wall is dominated by ejection motions. The solid and the dotted contour lines show contours of $\langle U\rangle =0$ and $\unicode[STIX]{x1D6FE}=90\,\%$ , respectively.

The $\langle u^{3}\rangle$ and $\langle u^{2}v\rangle$ distributions show that the dominant transport of turbulent kinetic energy is carried out by sweep motions ( $u>0$ and $v<0$ ) in the near wall area, and farther above the wall by the ejection motions ( $u<0$ , $v>0$ ). The boundary of the ejection and sweep motions overlaps with the local maximum of normal and shear Reynolds stresses in figure 6. It is conjectured that the ejection and sweep motions diffuse the local accumulation of the turbulent kinetic energy away from this area. The stronger sweep motions in the near-wall region agree with the observation of Krogstad & Skåre (Reference Krogstad and Skåre1995) and Lee & Sung (Reference Lee and Sung2008) regarding dominant turbulent transport toward the wall in the presence of APG. Song & Eaton (Reference Song and Eaton2002) also observed stronger sweep motions below the inflection point of the mean velocity profile in the separated flow region.