2 Experimental set-up

3 Results and discussion

Figure 6.

Time-averaged streamwise velocity $U/U_{0}$ contours in vertical $y$ – $z$ planes at $x/h=1$ , 4 and 7 and in the wall-parallel $x$ – $y$ plane at $z/h=0.41$ obtained from volumetric PIV measurements. Vectors indicate in-plane velocity field. (a,b) Trapezoidal tab, low and high $Re_{h}$ , respectively. (c,d) Triangular tab, low and high $Re_{h}$ , respectively.

Figure 7.

Time-averaged streamwise velocity $U/U_{0}$ profiles at the central plane of the four tabs at the low and high $Re_{h}$ . The results are obtained from high-spatial-resolution planar PIV. The velocity profiles are taken at the locations indicated in the abscissa and lines indicate rectangular (continuous), trapezoidal (dashed), triangular (dot-dashed) and ellipsoidal (long-dashed) tabs.

In this section we present the common and distinctive flow statistics and dominant vortical structures for the rectangular, trapezoidal, triangular and ellipsoidal tabs at the low and high $Re_{h}$ . Owing to the extensive experimental effort (four tab geometries, two Reynolds numbers and two measurement techniques) and for brevity, we highlight only selected representative cases without compromising the overall discussion. Although the tabs share various geometrical features, including base width, height and inclination angle, their different frontal geometries lead to distinctive characteristics and streamwise evolution of the vortical structures in the wake. First, the impact of the geometry and Reynolds number on the mean flow past the tabs is illustrated in figures 6–10. The normalized time-averaged streamwise velocity contours $U/U_{0}$ for the trapezoidal and triangular tabs are shown in figure 6 within the vertical $y$ – $z$ planes at $x/h=1$ , 4 and 7 and in the wall-parallel $x$ – $y$ plane at $z/h=0.41$ . In figure 6 and subsequent ones, the centre of the tab base is located at ( $0,0,0$ ), the velocity vectors indicate in-plane flow field and the wall-parallel plane at $z/h=0.41$ marks the lower boundary of the volumetric measurement region. For the two tabs shown in figure 6, and the other two which are not shown for brevity, the velocity distribution is dependent on the incoming flow regime (laminar versus turbulent), where the normalized deficit is reduced for each tab at the higher $Re_{h}$ (turbulent incoming flow). Furthermore, the larger frontal area of the trapezoidal tab (figure 6 a,b) compared to the triangular one (figure 6 c,d) leads, as expected, to a larger normalized flow deficit and near-wake size across both Reynolds numbers. This dependence on the flow-facing tab area was observed for the other tabs as well. The flow-facing tab areas as a percentage of that of the rectangular tab are 81 %, 79 % and 50 % for the trapezoidal, ellipsoidal and triangular tabs, respectively.

The $U/U_{0}$ profiles, shown in figure 7 and obtained using high-spatial-resolution planar PIV at the central plane of the four tabs, highlight the higher sensitivity of the flow field to the tab geometry at the low $Re_{h}$ . Only the rectangular tab exhibits flow reversal near the wall for ${\sim}3h$ downstream of the tab edge. The inflection points of the velocity profile differ in their vertical location across the tab geometry at the low $Re_{h}$ . Yang et al. (Reference Yang, Meng and Sheng2001) showed for a trapezoidal tab that the inflection point near $z/h=1$ is associated with the hairpin vortex head while the lower inflection point is associated with a secondary vortical structure with an opposite sense of rotation to the hairpin vortex head. Furthermore, the mean shear, $\unicode[STIX]{x2202}U/\unicode[STIX]{x2202}z$ , at the tab height is significantly enhanced at the low $Re_{h}$ , promoting KH instability, whose effects will be shown below. At the base of the tabs (figure 7), upstream flow reversal is observed at the low $Re_{h}$ , giving rise to the ‘necklace’ vortex that was first observed in the simulations of Dong & Meng (Reference Dong and Meng2004). However, this recirculation region is not present at the high  $Re_{h}$ .

Figure 8.

Time-averaged spanwise velocity $V/U_{0}$ contours for the trapezoidal tab at the low (a) and high (b) $Re_{h}$ in vertical $y$ – $z$ planes at $x/h=1$ , 4 and 7 and in the wall-parallel $x$ – $y$ plane at $z/h=0.41$ obtained from volumetric PIV measurements. Vectors indicate in-plane velocity field.

Figure 9.

Time-averaged wall-normal velocity $W/U_{0}$ contours for the trapezoidal tab at the low (a) and high (b) $Re_{h}$ in vertical $y$ – $z$ planes at $x/h=1$ , 4 and 7 and in the wall-parallel $x$ – $y$ plane at $z/h=0.41$ obtained from volumetric PIV measurements. Vectors indicate in-plane velocity field.

Figure 10.

Time-averaged wall-normal velocity $W/U_{0}$ profiles at $z/h=1$ (i.e. tab tip) for the four tabs at the low (a) and high (b) $Re_{h}$ . The results are obtained from the volumetric PIV measurements. Colours indicate profiles from the rectangular (green $+$ ), trapezoidal (red $\ast$ ), triangular (blue $\bigcirc$ ) and ellipsoidal (black $\times$ ) tabs.

Figure 11.

The counter-rotating vortex pair in the wake of the trapezoidal tab illustrated with $x$ -vorticity isosurfaces $\unicode[STIX]{x1D714}_{x}h/U_{0}=+0.1$ (red) and $-0.1$ (blue) at the low (a) and high (b) $Re_{h}$ . The $V/U_{0}$ contours and in-plane velocity vector fields in $y$ – $z$ planes at $x/h=3$ and 6 and in the $x$ – $y$ plane at $z/h=0.41$ are superimposed for clarity. The results are obtained from the volumetric PIV measurements.

Figure 12.

Streamwise evolution of the vertical location $z/h$ of the CVP core averaged across both branches at the low (a) and high (b) $Re_{h}$ . Results are from the volumetric PIV measurements for the rectangular (green $+$ ), trapezoidal (red $\ast$ ), triangular (blue $\bigcirc$ ) and ellipsoidal (black $\times$ ) tabs.

Figure 13.

Streamwise evolution of the normalized circulation $\unicode[STIX]{x1D6E4}/\unicode[STIX]{x1D6E4}_{0}$ averaged across both branches of the CVP at the low (a) and high (b) $Re_{h}$ . Results are from the volumetric PIV measurements for the rectangular (green $+$ ), trapezoidal (red $\ast$ ), triangular (blue $\bigcirc$ ) and ellipsoidal (black $\times$ ) tabs.

The normalized time-averaged spanwise $V/U_{0}$ and wall-normal $W/U_{0}$ velocity distributions for the trapezoidal tab at both $Re_{h}$ , shown in figures 8 and 9, reveal the signature of the CVP that induces significant mean spanwise flow, and upward wall-normal entrainment at the tab symmetry plane. The spanwise flow $V$ reaches up to ${\sim}0.17U_{0}$ at the low $Re_{h}$ and exhibits a decrease in the overall distribution at the high $Re_{h}$ (figure 8). An upward flow is formed in the region between the two branches of the CVP and reaches up to ${\sim}0.15U_{0}$ at the symmetry plane at the low $Re_{h}$ (figure 9). This upward flow is essential in the mixing and transport of near-wall fluid into the free-stream flow (Dong & Meng Reference Dong and Meng2004). Similar to the spanwise component, $W/U_{0}$ is significantly lower at the high $Re_{h}$ . The reduction in $V/U_{0}$ and $W/U_{0}$ at the high $Re_{h}$ was also observed for the other tabs. To further inspect the effect of Reynolds number and tab geometry on the vertical flow, figure 10 shows one-dimensional (1D) profiles of $W/U_{0}$ at $z/h=1$ (tab tip) for the four tabs at both $Re_{h}$ . Despite some spanwise asymmetry, probably a result of multiple factors, fair comparison across the different tab geometries is provided from these profiles. Upstream of $x/h=2$ , the region of enhanced vertical flow is mainly due to the inclined geometry of the tabs. Beyond $x/h=2$ , the enhanced vertical flow is due to the common-up entrainment induced by the CVP and the upward pumping by intermittent hairpin vortices. These two primary vortical structures are discussed in detail below. Overall, the vertical flow is reduced at the high $Re_{h}$ , with the triangular tab showing the lowest wall-normal flow across both $Re_{h}$ . At the low $Re_{h}$ , the rectangular and trapezoidal tabs maintain higher wall-normal flow; while at the high $Re_{h}$ , the rectangular tab performs better than the other tabs.

The 3D topology of the CVP, which forms due to the pressure drop across the tabs (Dong & Meng Reference Dong and Meng2004), is illustrated for the trapezoidal tab at both $Re_{h}$ in figure 11 through isosurfaces of the normalized $x$ -vorticity $\unicode[STIX]{x1D714}_{x}h/U_{0}=\pm 0.1$ with superimposed $V/U_{0}$ planar contours. In contrast to the hairpin vortices, the CVP is present in the mean flow field and, as such, $\unicode[STIX]{x1D714}_{x}$ is here calculated using the time-averaged flow field ( $V,W$ ). The structures in figure 11 were also visualized using the $Q$ (Hunt, Wray & Moin Reference Hunt, Wray and Moin1988) and $\unicode[STIX]{x1D706}_{2}$ (Jeong & Hussain Reference Jeong and Hussain1995) criteria and showed similar topology. It is noted that, at the low $Re_{h}$ , a secondary CVP with an opposite sense of pair rotation to that of the primary one is observed downstream of $x/h=5$ (figure 11 a). This secondary structure was observed in the simulations of Dong & Meng (Reference Dong and Meng2004) and Habchi et al. (Reference Habchi, Lemenand, Valle and Peerhossaini2010b ) and was identified as detrimental to mixing, as it induces a downward flow in the symmetry axis of the tab. We provide experimental evidence of this secondary structure appearing at the low $Re_{h}$ ; however, this structure was not observed at the high $Re_{h}$ . In fact, this structure was observed for all tabs at the low $Re_{h}$ but for none at the high $Re_{h}$ . It is likely that the more prominent wall vortical structures at the high $Re$ result in reducing the coherence of this near-wall secondary CVP. Additional measurements of higher resolution are needed to investigate the presence of this structure. As seen in figure 11, the primary CVP is inclined at a higher angle at the low $Re_{h}$ , possibly due to the presence of the secondary CVP and the lower advection at the low $Re_{h}$ . Following a procedure similar to that of Lögdberg, Fransson & Alfredsson (Reference Lögdberg, Fransson and Alfredsson2009) and Habchi et al. (Reference Habchi, Lemenand, Valle and Peerhossaini2010b ), the location of the core of the CVP is identified as the point of maximum $Q_{x}$ , where

is the second invariant of the 2D velocity gradient tensor calculated locally in a given $y$ – $z$ plane. In figure 12, the height of the core for both branches is averaged and shows steeper slope at the low Reynolds number across all tabs.

Further, the circulation

is calculated locally at each $y$ – $z$ plane. Here, $|\unicode[STIX]{x1D714}_{x}|h/U_{0}$ is the normalized vorticity in the $x$ -direction and $A$ is the normalized area of the vortex defined as the region where $Q_{x}$ drops to 5 % of its maximum value at the core (Lögdberg et al. Reference Lögdberg, Fransson and Alfredsson2009; Habchi et al. Reference Habchi, Lemenand, Valle and Peerhossaini2010b ). Figure 13 shows the circulation for all tabs at the two $Re_{h}$ averaged across both branches and normalized by the global maximum, $\unicode[STIX]{x1D6E4}_{0}$ , which occurs for the trapezoidal tab at $x/h\sim 4$ at the low $Re_{h}$ . Larger circulation and slower decay are observed at the low $Re_{h}$ . This is attributed to the reduced background turbulence from the incoming boundary layer flow (shown in figure 17), which reduces the coherence of the vortical structures induced by the tabs. Consequently, the reduced coherence and circulation at the high $Re_{h}$ lead to the lower mean spanwise and wall-normal flows discussed earlier. The strength of the CVP exhibits distinctive streamwise evolution across Reynolds number. At the low $Re_{h}$ , continuous growth in $\unicode[STIX]{x1D6E4}/\unicode[STIX]{x1D6E4}_{0}$ is observed until a maximum occurs at $x/h\sim 4$ , while the maximum is reached significantly earlier at $x/h\sim 2.5$ at the high $Re_{h}$ . The lower incoming flow turbulence at the low $Re_{h}$ allows for the coherence and strength of the CVP to be preserved for a larger downstream distance. Owing to the important role of the CVP in mixing, the distinctive behaviour and topology across $Re_{h}$ suggests higher efficiency through CVP mixing at the low $Re_{h}$ . Moreover, when such tabs are placed in rows, the spacing must be chosen carefully in correspondence with the operating $Re_{h}$ to maintain energetic steady mixing. Given that the CVP originates due to the pressure drop, it is expected that tab blockage affects the strength of the CVP. However, as seen in figure 13, the normalized circulation does not directly correspond to blockage and is shown to be highly dependent on geometry and incoming flow regime. This suggests the need for application-specific testing when choosing a tab geometry where mixing by the CVP is desired.

Quasi-periodic hairpin vortices in the wake of trapezoidal tabs have been reported through 2D flow visualizations and planar PIV (Elavarasan & Meng Reference Elavarasan and Meng2000; Yang et al. Reference Yang, Meng and Sheng2001). However, the 2D character of these measurements falls short in its ability to describe the 3D topology of the hairpin vortices and their interaction with the CVP. Although limited to a low $Re_{h}=600$ , the DNS simulations by Dong & Meng (Reference Dong and Meng2004) have greatly advanced the understanding of these structures. They observed that the hairpin vortex heads are rolled from the shear layer formed at the trailing edge of the tab. Moreover, they suggested splitting and deformation as the mechanism responsible for transforming the CVP into the hairpin vortex legs. Our volumetric PIV measurements show the hairpin vortices and their interaction with the CVP at a $Re_{h}$ one order of magnitude higher than that of Dong & Meng (Reference Dong and Meng2004). Furthermore, our measurements show that the CVP does not necessarily evolve into the hairpin legs, and that there are various forms of interaction. Figure 14 illustrates a representative instant where the coexistence of the hairpin structures and the CVP takes place in the wake of the ellipsoidal tab at $Re_{h}=2000$ . There, the hairpin structures are identified using an isosurface of the normalized vorticity magnitude $|\unicode[STIX]{x1D714}|h/U_{0}=1.2$ . Two stream traces (with $u-u_{c}$ , $v$ , $w$ , where $u_{c}$ is the velocity at the core of the downstream hairpin head) are shown to highlight the flow direction and circulation at the hairpin head. The instantaneous CVP (in contrast to the mean one in figure 11) is identified using an isosurface of the normalized $Q$ criterion $Q_{x}h^{2}/U_{0}^{2}=0.02$ , where $Q_{x}$ is calculated at consecutive $y$ – $z$ planes using (3.1). Note that, for figure 14, both $\unicode[STIX]{x1D714}$ and $Q_{x}$ are calculated based on the instantaneous flow field ( $u,v,w$ ). While hairpin vortices were observed for all tabs, we illustrate here those in the wake of an ellipsoidal tab to show that a flat trailing edge is not a necessary condition for the generation of hairpin vortices. This is in contrast with the work of Gretta (Reference Gretta1990), who suggested the need for a flat tab trailing edge for the generation of hairpin vortices. However, we point out that a flat edge is likely to enhance the shear, $\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}z$ , leading to stronger hairpin vortices, as discussed in the context of the 2D measurements below. As seen in figure 14, hairpin vortices coexist and interact with the CVP. Although the CVP is influenced and deformed due to this interaction, it remains present, in contrast with the observations of Dong & Meng (Reference Dong and Meng2004). An enlarged view of the region $6\leqslant x/h\leqslant 7$ is given in figure 15 and suggests at least two forms of interaction between the hairpin vortex legs and CVP structures. The region A (figure 15 a) highlights the coexistence, and not merging, of the hairpin leg with the CVP. The branch of the CVP with negative streamwise vorticity continues past the hairpin head, while the branch with positive streamwise vorticity undergoes deformation and splits around the hairpin head, as seen in region B (figure 15 b). Similar instantaneous interactions have been observed for the other tabs. High-frame-rate measurements are needed to further investigate the temporal behaviour of these large-scale vortices. Such measurements would allow for a better description of the interaction of the CVP with the hairpin vortices.

Figure 14.

Coexistence of hairpin vortices with CVP in the wake of an ellipsoidal tab at the low $Re_{h}$ . Hairpin structures (in blue) are identified using the $|\unicode[STIX]{x1D714}|h/U_{0}=1.2$ isosurface. CVP (in red) is identified using the $Q_{x}h^{2}/U_{0}^{2}=0.02$ isosurface. Two stream traces (with $u-u_{c}$ , $v$ , $w$ , where $u_{c}$ is the velocity at the core of the hairpin head) are shown to depict the sense of rotation. The levels for the $u/U_{0}$ contours in the wall-parallel $x$ – $y$ plane at $z/h=0.41$ are the same as in figure 6. The results are obtained from volumetric PIV measurements; the hairpin vortices without the CVP are shown in the top right corner for clarity.

Figure 15.

Enlarged view from figure 14 around $6\lesssim x/h\lesssim 7$ illustrating the interaction between the hairpin vortex and CVP.

Further insight into the formation, dynamics and trajectory of the heads of the hairpin vortices is gained from the representative instantaneous signed swirling strength $\unicode[STIX]{x1D6EC}_{ci}h/U_{0}$ fields obtained from the 2D high-resolution measurements given in figure 16. Here $\unicode[STIX]{x1D6EC}_{ci}$ is the magnitude of the imaginary part of the complex eigenvalues of the velocity gradient tensor (Zhou et al. Reference Zhou, Adrian, Balachandar and Kendall1999) and its sign indicates the orientation of the swirl (Wu & Christensen Reference Wu and Christensen2006). Figure 16 highlights the generation of hairpin vortices (heads) in the case of triangular and ellipsoidal tabs regardless of their lack of a flat trailing edge. Moreover, figure 16 contrasts the dynamics across $Re_{h}$ for the four tabs. In particular, the necklace vortex at the upstream edge of the four tabs, illustrated at the top left corner of the rectangular tab at low $Re_{h}$ , is not present at the high $Re_{h}$ . At the low $Re_{h}$ , the high shear, $\unicode[STIX]{x2202}u/\unicode[STIX]{x2202}z$ , promotes KH instability that manifests in the periodic shedding of hairpin vortices. In fact, the flow is mostly laminar upstream of this location at this $Re_{h}$ . Interestingly, the geometry affects the location of the onset of hairpin vortices at the low $Re_{h}$ . A careful review of numerous instantaneous fields suggests that the largest delay in hairpin vortex location occurs for triangular tabs. The vortex dynamics at the high $Re_{h}$ are remarkably richer due to the turbulent nature of the incoming and wake flows. The entire wake exhibits a high population of vortical structures and lacks the well-formed KH instability observed at the low $Re_{h}$ . The multi-scale finer structures around the large-scale hairpin vortices affect their strength, path and coherence.

Figure 16.

Instantaneous signed swirling strength $\unicode[STIX]{x1D6EC}_{ci}h/U_{0}$ at the central plane of the four tabs at both $Re_{h}$ . The top left panel shows an example of the necklace vortex formed at the leading edge of the tabs for the low $Re$ cases. The results are obtained from high-spatial-resolution planar PIV measurements.

Figure 17.

Turbulent kinetic energy $\text{TKE}=\langle {u^{\prime }}^{2}+{w^{\prime }}^{2}\rangle /2U_{0}^{2}$ fields for the four tabs at both $Re_{h}$ . The results are obtained from the high-spatial-resolution planar PIV measurements.

Figure 18.

Primary Reynolds shear stress $-\langle u^{\prime }w^{\prime }\rangle /U_{0}^{2}$ fields for the four tabs at both $Re_{h}$ . The results are obtained from the high-spatial-resolution planar PIV measurements.

Table 2.

Normalized average circulation $\bar{\unicode[STIX]{x1D6E4}}/(U_{0}h)$ and vortex core radius $\bar{r_{c}}/h$ of hairpin heads observed within $5.5\leqslant x/h\leqslant 7.5$ for the four tabs at the low $Re_{h}$ . Subindex $r$ indicates a quantity from the rectangular tab.

The comparatively low turbulence levels at the low $Re_{h}$ allow for further comparisons in the strength and scale of the hairpin heads across tab geometries. Following Moriconi (Reference Moriconi2009), we chose the Lamb–Oseen vortex as a simplified model to represent the head of hairpin vortices identified at the low $Re_{h}$ in figure 16. The Lamb–Oseen vortex is a non-stationary solution of the Navier–Stokes equations given in cylindrical coordinates as

Here, $r_{c}$ is the vortex core radius at a given instant and $r$ is the radial distance from the vortex core. Although a simplified model, it shows remarkable agreement with the measured data. To allow for comparisons across different tabs, we compute the normalized mean circulation $\bar{\unicode[STIX]{x1D6E4}}/(U_{0}h)$ and vortex core radius $\bar{r_{c}}/h$ for hairpin heads within $5.5\leqslant x/h\leqslant 7.5$ across 30 instantaneous fields. Both $\unicode[STIX]{x1D6E4}$ and $r_{c}$ were obtained from curve fitting individual hairpin heads to the Lamb–Oseen model; $\bar{\unicode[STIX]{x1D6E4}}/(U_{0}h)$ and $\bar{r_{c}}/h$ are reported in table 2. The rectangular tab has the highest circulation and core radius, while the triangular tab shows the weaker and smaller hairpin heads. To facilitate comparison, the quantities for the other tabs are scaled by that of the rectangular. There, it is noted that the circulation of the hairpin heads for the triangular tab are ${\sim}50\,\%$ of those of the rectangular one, but their size is similar ( $r_{c}$ for the triangular tab is $92\,\%$ of the rectangular). As reported in table 2, tab geometry, through blockage and trailing edge, plays a significant role in determining the scale and strength of the generated hairpin vortices.

Finally, the turbulent kinetic energy $\text{TKE}=\langle {u^{\prime }}^{2}+{w^{\prime }}^{2}\rangle /2U_{0}^{2}$ and primary Reynolds shear stress $-\langle u^{\prime }w^{\prime }\rangle /U_{0}^{2}$ obtained from the planar PIV measurements are given in figures 17 and 18 for the four tabs at both $Re_{h}$ . The figures illustrate the distinctive behaviour of the velocity fluctuations across $Re_{h}$ . In contrast to the high $Re_{h}$ case, enhanced turbulence levels are delayed by ${\sim}(2{-}3)h$ behind the downstream edge of the tab. In this region, the high shear promotes KH instability. The reduced turbulence levels within this region along with the delayed generation of hairpin vortices (figure 16) allow the CVP to maintain coherence and growth in strength in contrast to the high $Re_{h}$ case (figure 13). Figures 17 and 18 suggest that this instability develops over a larger distance for the triangular tab, which is in line with the mean shear, $\unicode[STIX]{x2202}U/\unicode[STIX]{x2202}z$ , inferred from figure 7. Dong & Meng (Reference Dong and Meng2004) and Habchi et al. (Reference Habchi, Lemenand, Valle and Peerhossaini2010b ) have discussed in detail the role of the various vortical structures in mixing. Given the distinctive vortex dynamics across $Re_{h}$ and tab geometry, the problem of mixing maximization proves to be a complex one best determined at the operational $Re_{h}$ .