3.3.1 Actuation $S_{1}$ along the upper edge of the rear window

The dependence of $C_{D}$ and $\unicode[STIX]{x0394}C_{D}$ on $C_{\unicode[STIX]{x1D707}}^{S1}$ is given in figure 10 as $S_{1}$ is operated at the blowing angles $\unicode[STIX]{x1D703}_{S1}=0^{\circ }$ , $30^{\circ }$ , $65^{\circ }$ and $90^{\circ }$ . $C_{D}$ and $\unicode[STIX]{x0394}C_{D}$ are apparently correlated to each other. There appear two critical momentum coefficients, $C_{\unicode[STIX]{x1D707},c1}^{S1}$ and $C_{\unicode[STIX]{x1D707},c2}^{S1}$ , for each $\unicode[STIX]{x1D703}_{S1}$ . $C_{D}$ rises initially with increasing $C_{\unicode[STIX]{x1D707}}^{S1}$ given $C_{\unicode[STIX]{x1D707}}^{S1}\leqslant C_{\unicode[STIX]{x1D707},c1}^{S1}$ , due to an increased separation region over the slanted surface at small $C_{\unicode[STIX]{x1D707}}^{S1}$ , which results in a pressure drop over the slanted surface (Aubrun et al. Reference Aubrun, Mcnally, Alvi and Kourta2011). Once $C_{\unicode[STIX]{x1D707}}^{S1}$ exceeds $C_{\unicode[STIX]{x1D707},c1}^{S1}$ , $C_{D}$ drops, reaching the minimum at $C_{\unicode[STIX]{x1D707}}^{S1}=C_{\unicode[STIX]{x1D707},c2}^{S1}$ before rising again. The value of $C_{\unicode[STIX]{x1D707},c2}^{S1}$ increases from $0.5\times 10^{-2}$ to $1.6\times 10^{-2}$ from $\unicode[STIX]{x1D703}_{S1}=0^{\circ }$ to $90^{\circ }$ ; meanwhile, the maximum DRs (figure 10 b) are approximately 3 %, 10 %, 12 % and 3 % for $\unicode[STIX]{x1D703}_{S1}=0^{\circ }$ , $30^{\circ }$ , $65^{\circ }$ and $90^{\circ }$ , respectively. A $\unicode[STIX]{x0394}C_{D}$ of $-10\,\%$ corresponds to a decrease in $C_{D}$ of approximately 0.036 for $Re=1.7\times 10^{5}$ . This result suggests that the optimal control is achieved at $\unicode[STIX]{x1D703}_{S1}=65^{\circ }$ , i.e. a blowing angle normal to the slanted surface. It will be shown later that the $S_{1}$ operated at this  $\unicode[STIX]{x1D703}_{S1}$ produces a considerable alteration in the flow separation over the rear window and the upper and lower recirculation bubbles behind the base, accounting for the significant DR.

Figure 10. Dependence on $C_{\unicode[STIX]{x1D707}}^{S1}$ of (a) $C_{D}$ and (b) $\unicode[STIX]{x0394}C_{D}$ under $S_{1}$ at $\unicode[STIX]{x1D703}_{S1}=0^{\circ }$ , $30^{\circ }$ , $65^{\circ }$ and $90^{\circ }$ for $Re=1.7\times 10^{5}$ . The uncertainty bars of $\unicode[STIX]{x0394}C_{D}$ are determined as the ratio of the uncertainty bars of $C_{D}$ to $\overline{\overline{C}_{D0}}$ . The vertical broken lines indicate the critical momentum coefficients.

Figure 11. (a) Variation in $C_{p}$ on the rear window and the vertical base at $y^{\ast }=0$ and 0.45, and change in $\langle C_{p}\rangle$ , under $S_{1}$ ( $\unicode[STIX]{x1D703}_{S1}=65^{\circ }$ ) at $C_{\unicode[STIX]{x1D707}}^{S1}=1.1\times 10^{-2}$ , and the corresponding time-averaged flow structure (b) in the ( $x$ , $z$ ) plane of $y^{\ast }=0$ and (c) in the ( $y$ , $z$ ) plane of $x^{\ast }=0.2$ . The uncertainty bars of $C_{p}$ in (a) are computed by using $\overline{\overline{|C_{p}-\overline{\overline{C_{p}}}|}}$ . Red-coloured symbols ‘ $\times$ ’ and ‘ $+$ ’ in (b,c) denote the saddle point and the focus, respectively, which are determined from time-averaged streamlines in figures 7(b) and 8(b). The red-coloured broken line in (b) indicates the bubble size in the base flow determined from the streamlines in figure 7(b). The contour interval and cutoff levels are 1 and $\pm 1$ , respectively, for $\overline{\unicode[STIX]{x1D714}_{x}^{\ast }}$ in (c). $Re=1.7\times 10^{5}$ .

The variations in $C_{p}$ on the rear window and the vertical base are compared in the vertical planes of $y^{\ast }=0$ and 0.45 (figure 11 a), which may provide insight into the flow physics behind the observed DR. In the measurement of $C_{p}$ , the actuators were present in the uncontrolled flow, although not operated. The uncertainty of $C_{p}$ is estimated to be within 0.01, corresponding to approximately 2 % of the spatially averaged pressure coefficient $(-0.43)$ of $C_{p2}$ – $C_{p19}$ . This averaged pressure is in good agreement with Lienhart & Becker’s (Reference Lienhart and Becker2003) estimate $(-0.41)$ for an Ahmed body ( $\unicode[STIX]{x1D711}=25^{\circ }$ , $Re=8.9\times 10^{5}$ ). When $S_{1}$ is operated at $C_{\unicode[STIX]{x1D707}}^{S1}=1.1\times 10^{-2}$ ( $\unicode[STIX]{x1D703}_{S1}=65^{\circ }$ ), where a maximum DR of 12 % is achieved, $C_{p2}$ at $z^{\ast }=0.81$ ( $y^{\ast }=0$ ) rises by 43 %. However, the rise is rather mild downstream for $C_{p3}$ (17 %) and $C_{p4}$ (6 %), and even becomes indiscernible when approaching $C_{p5}$ . At $y^{\ast }=0.45$ , $C_{p11}$ , $C_{p12}$ , $C_{p13}$ and $C_{p14}$ are increased by approximately 31 %, 13 %, 8 % and 4 %, respectively, as compared with their uncontrolled counterparts. Under control, flow separation is postponed downstream from the upper edge of the rear window, as will be shown later from the PIV data. This alternation is responsible for the overall recovery over the slanted surface. The values of $C_{p6}$ – $C_{p10}$ in the symmetry plane of the base increase but $C_{p15}$ – $C_{p19}$ near the side edge drop, which are all linked to a change in the upper and lower recirculation bubbles, as will be shown later. An overall increase in the spatially averaged pressure by 12 % is produced under $S_{1}$ . It is worth pointing out that the variation in the averaged pressure is always approximately the same as the DR under control, for $S_{1}$ , $S_{2}$ , $S_{3}$ , $S_{4}$ or their combinations.

$C_{p1}$ measured at the centre of the front face is approximately 1.0 with and without control. Similar observations have also been made for actuations $S_{2}$ , $S_{3}$ and $S_{4}$ . The altered time-averaged flow structures under $S_{1}$ are examined based on PIV measurements in an effort to unveil the DR mechanisms. Figure 11(b) presents the time-averaged streamlines measured in the symmetry plane under $S_{1}$ of $\unicode[STIX]{x1D703}_{S1}=65^{\circ }$ at the optimal $C_{\unicode[STIX]{x1D707}}^{S1}\;(=1.1\times 10^{-2})$ . A number of observations can be made from a comparison between the flows with and without control (figure 7 b).

Firstly, flow separation under $S_{1}$ does not occur at the upper edge of the rear window anymore and is instead shifted downstream to $x^{\ast }\approx -0.24$ , which is highlighted by a thick contour circle, in distinct contrast to the base flow (figure 7 b) where the flow remains attached over the rear window. This is not unexpected since the microjet array of $S_{1}$ generates streamwise vortices (e.g. Kumar & Alvi Reference Kumar and Alvi2006). These vortices entrain higher momentum free-stream fluid to the near-wall low momentum region, which energizes the boundary layer fluid and prevents or delays flow separation. Furthermore, a separated shear layer appears joining the upper recirculation bubble behind the vertical base, that is, the flow separation region over the rear window becomes part of the recirculation bubble behind the base. As a result, there is a moderate pressure rise, by 6 %, in $C_{p4}$ at the centre of the rear window (figure 11 a).

The upper and lower recirculation bubbles are enlarged longitudinally under $S_{1}$ (figure 11 b). To facilitate data interpretation, the bubbles, centres and saddle points shown in figure 7(b) are indicated by a broken curve and symbols ‘ $+$ ’ and ‘ $\times$ ’ in red colour, respectively. The centre of the upper recirculation bubble under control appears shifting downstream and upward, relative to the base flow. The centre of the lower recirculation bubble and the saddle point also shift downstream. The sizes and centres of the recirculation bubbles are found to be closely connected to the surface pressure on the vertical base and hence the drag of the model (e.g. Pastoor et al. Reference Pastoor, Henning, Noack, King and Tadmor2008; Metka & Gregory Reference Metka and Gregory2015; Rossitto et al. Reference Rossitto, Sicot, Ferrand, Borée and Harambat2016). This connection is evident in the variation of $C_{p}$ on the vertical base (figure 11 a). The value of $C_{p}$ displays a minimum, $-0.32$ , at the position of $C_{p8}$ ( $z^{\ast }=0.29$ ) in the uncontrolled flow. This minimum occurs at nearly the same height as that ( $z^{\ast }=0.27$ ) of the upper recirculation bubble centre (figure 11 b), apparently resulting from its proximity to this centre. The centre of the upper bubble moves downstream from $x^{\ast }=0.12$ to 0.38 under control, causing a rise in $C_{p8}$ by 22 %. At the same time, this centre goes up from $z^{\ast }=0.27$ to 0.41 (figure 11 b); the minimum $C_{p}$ on the base moves to $C_{p6}$ at $z^{\ast }=0.41$ (figure 11 a), again at the same height as the bubble centre. This observation reinforces the concept that the location of the recirculation bubble centre has a considerable effect on the base pressure.

In view of the highly three-dimensional (3-D) flow, the altered flow structure under control is examined in the ( $y$ , $z$ ) plane of $x^{\ast }=0.2$ . Under $S_{1}$ , the maximum concentration of $\overline{\unicode[STIX]{x1D714}_{x}^{\ast }}$ on each side of the rear window (figure 11 c), which corresponds to the C-pillar vortex marked by ‘C’, reaches approximately 12, almost the same as that (at 13) in the base flow (figure 8 c). This result indicates that the actuation fails to weaken appreciably the C-pillar vortices. Further, the time-averaged streamlines (figure 11 c) display two foci at ( $y^{\ast }$ , $z^{\ast }$ )  $=$  ( $-0.38$ , 0.35) and ( $-0.45$ , 0.12), which are connected to the legs of the upper and lower recirculation bubbles, respectively. It appears that the upper and lower recirculation bubbles under $S_{1}$ tilt upstream toward the vertical base near the side edge of the base, as suggested by the anti-clockwise and clockwise rotational motions around the upper and lower foci, respectively. In comparison, the streamlines in the ( $y$ , $z$ ) plane of $x^{\ast }=0.2$ show only one pair of foci at $y^{\ast }=\pm 0.22$ and $z^{\ast }=0.25$ behind the base in the uncontrolled flow (figure 8 b), corresponding to the trailing legs of the upper recirculation bubble. The modifications in the two bubbles account for the drop in $C_{p15}$ – $C_{p19}$ near the side edge of the base (figure 11 a).

The effectiveness of a steady slot jet (0.8 mm in width and 138 mm in length) arranged at the same position as $S_{1}$ is also examined. This actuation is referred to as $S_{1}^{slot}$ . The blowing direction is normal to the rear window, the same as $S_{1}$ ( $\unicode[STIX]{x1D703}_{S1}=65^{\circ }$ ). Such actuation leads to an increase in drag by 11 % (not shown), in distinct contrast to the maximum DR of 12 % under $S_{1}$ (figure 10 b). The difference between $S_{1}^{slot}$ and $S_{1}$ is ascribed to a difference in the jet exit shape. The rectangular slot produces a quasi-2-D blowing jet, while the microjet array generates 3-D streamwise vortices (e.g. Johnston & Nishi Reference Johnston and Nishi1990; Yang & Zhou Reference Yang and Zhou2016). The streamwise vortices enhance cross-stream mixing of streamwise momentum and hence suppress flow separation from the upper edge of the rear window, as supported by the PIV data (figure 11 b).

3.3.2 Actuation $S_{2}$ along the two side edges of the rear window

As $S_{2}$ is operated at $\unicode[STIX]{x1D703}_{S2}=30^{\circ }$ , $C_{D}$ displays a slight drop initially with increasing $C_{\unicode[STIX]{x1D707}}^{S2}$ but grows continuously once $C_{\unicode[STIX]{x1D707}}^{S2}$ exceeds $0.2\times 10^{-2}$ (figure 12). The $C_{D}$ drops slowly at $\unicode[STIX]{x1D703}_{S2}=60^{\circ }$ with increasing $C_{\unicode[STIX]{x1D707}}^{S2}$ but more rapidly at $\unicode[STIX]{x1D703}_{S2}=90^{\circ }$ , with $\unicode[STIX]{x0394}C_{D}$ reaching $-6\,\%$ at $C_{\unicode[STIX]{x1D707}}^{S2}=0.7\times 10^{-2}$ . This observation implies that $S_{2}$ may be more effective in increasing the swirl of the C-pillar vortices when directed normally to the slanted surface. The increasing swirl may act to propel the C-pillar vortices away to become diffuse, causing a decrease in their strength (Bruneau et al. Reference Bruneau, Creuse, Delphine, Gilliéron and Mortazavi2011).

Figure 12. Dependence on $C_{\unicode[STIX]{x1D707}}^{S2}$ of (a) $C_{D}$ and (b) $\unicode[STIX]{x0394}C_{D}$ under $S_{2}$ at $\unicode[STIX]{x1D703}_{S2}=30^{\circ }$ , $60^{\circ }$ and $90^{\circ }$ for $Re=1.7\times 10^{5}$ .

The deployment of $S_{2}$ ( $\unicode[STIX]{x1D703}_{S2}=90^{\circ }$ ) does not produce any marked change in the surface pressure over the rear window or at the vertical base. At $C_{\unicode[STIX]{x1D707}}^{S2}=0.7\times 10^{-2}$ , the highest magnitude of the $\overline{\unicode[STIX]{x1D714}_{x}^{\ast }}$ concentration for the C-pillar vortices in the PIV measured $\overline{\unicode[STIX]{x1D714}_{x}^{\ast }}$ -contours in the ( $y$ , $z$ ) plane of $x^{\ast }=0.2$ is reduced to 10 (not shown), compared with that (13) in the base flow (figure 8 c). As a result, $C_{p11}$ , $C_{p12}$ , $C_{p13}$ and $C_{p14}$ near the side edge of the rear window under control rise by 8 %, 7 %, 4 % and 4 %, respectively, which are ascribed to the weakened strength of the C-pillar vortices. On the other hand, the $C_{p2}$ – $C_{p5}$ in the symmetry plane of the slanted surface, and $C_{p6}$ – $C_{p10}$ and $C_{p15}$ – $C_{p21}$ at the base, change little under control.

It is worth mentioning that two arrays of steady circular microjets, denoted $S_{2}^{microjet}$ , directed normally to the slanted surface were used to replace $S_{2}$ . The exit diameter of and separation between the orifices are the same as $S_{1}$ . The maximum DR obtained is very small, only approximately 1 % (not shown), that is, the slot jets are more effective than the microjet arrays in manipulating the C-pillar vortices.

3.3.3 Actuation $S_{3}$ along the upper edge of the vertical base

The effect of $S_{3}$ on $\unicode[STIX]{x0394}C_{D}$ is examined for five different blowing angles $\unicode[STIX]{x1D703}_{S3}$ . At $\unicode[STIX]{x1D703}_{S3}>0$ , $C_{D}$ decreases with increasing $C_{\unicode[STIX]{x1D707}}^{S3}$ (figure 13 a), $\unicode[STIX]{x0394}C_{D}$ reaching $-11\,\%$ and $-7\,\%$ at $C_{\unicode[STIX]{x1D707}}^{S3}=6.9\times 10^{-2}$ for $\unicode[STIX]{x1D703}_{S3}=45^{\circ }$ and $30^{\circ }$ (figure 13 b), respectively. At $\unicode[STIX]{x1D703}_{S3}=0^{\circ }$ , the drag is slightly increased. For $\unicode[STIX]{x1D703}_{S3}<0^{\circ }$ , however, $\unicode[STIX]{x0394}C_{D}$ continuously rises with increasing $C_{\unicode[STIX]{x1D707}}^{S3}$ , reaching 11 % and 20 % at $\unicode[STIX]{x1D703}_{S3}=-30^{\circ }$ and $-45^{\circ }$ ( $C_{\unicode[STIX]{x1D707}}^{S3}=6.9\times 10^{-2}$ ), respectively.

Figure 13. Dependence on $C_{\unicode[STIX]{x1D707}}^{S3}$ of (a) $C_{D}$ and (b) $\unicode[STIX]{x0394}C_{D}$ under $S_{3}$ at $\unicode[STIX]{x1D703}_{S3}=45^{\circ }$ , $30^{\circ }$ , $0^{\circ }$ , $-30^{\circ }$ and $-45^{\circ }$ for $Re=1.7\times 10^{5}$ .

At $C_{\unicode[STIX]{x1D707}}^{S3}=6.9\times 10^{-2}$ ( $\unicode[STIX]{x1D703}_{S3}=45^{\circ }$ ) where the maximum DR of 11 % is produced, $C_{p2}$ increases by 27 % in the plane of symmetry (figure 14 a). The pressure rise contracts downstream, $\unicode[STIX]{x0394}C_{p3}$ , $\unicode[STIX]{x0394}C_{p4}$ and $\unicode[STIX]{x0394}C_{p5}$ being 10 %, 4 % and 2 %, respectively. Similar observation is made near the side edge of the rear window, where $C_{p11}$ , $C_{p12}$ , $C_{p13}$ and $C_{p14}$ rise by 25 %, 5 %, 4 % and 4 %, respectively. It is interesting to note that $S_{3}$ , although deployed along the upper edge of the vertical base, has a great influence on the surface pressure over the rear window, implying an upstream effect on the flow structure over the slanted surface. $S_{3}$ acts to increase $C_{p6}$ – $C_{p10}$ in the symmetry plane of the base on one hand but to decrease $C_{p15}$ – $C_{p19}$ near the side edge on the other hand. It will be later shown that $S_{3}$ incurs a marked change in the flow structure over the rear window and behind the vertical base, which accounts for the pressure variation.

Figure 14. (a) Variation in $C_{p}$ on the rear window and the vertical base at $y^{\ast }=0$ and 0.45, and change in $\langle C_{p}\rangle$ , under $S_{3}$ ( $\unicode[STIX]{x1D703}_{S3}=45^{\circ }$ ) at $C_{\unicode[STIX]{x1D707}}^{S3}=6.9\times 10^{-2}$ , and the corresponding time-averaged flow structure (b) in the ( $x$ , $z$ ) plane of $y^{\ast }=0$ and (c) in the ( $y$ , $z$ ) plane of $x^{\ast }=0.2$ . The symbols ‘ $\times$ ’ and ‘ $+$ ’ and the red-coloured broken line in (b,c) are as in figure 11(b,c). The contour interval and cutoff levels for $\overline{\unicode[STIX]{x1D714}_{x}^{\ast }}$ in (c) are the same as in figure 11(c). $Re=1.7\times 10^{5}$ .

Under this control, flow separation occurs downstream of the upper edge of the rear window and the separated flow reattaches on the slanted surface, forming a separation bubble (figure 14 b). Its size is much larger than that of the small separation bubble formed near the upper edge of the rear window in the base flow (figure 7 b). Furthermore, the separated shear layer from the upper edge of the base is deflected upwards by $S_{3}$ , and the upper and lower recirculation bubbles are longitudinally more stretched than the non-controlled case. Meanwhile, the centres of the two bubbles, as well as the saddle point, are pushed downstream. It is therefore plausible that the pressure rise of $C_{p8}$ at the centre of the base by 31 % is due to the modifications in the upper and lower recirculation bubbles behind the base.

The flow structure in the ( $y$ , $z$ ) plane of $x^{\ast }=0.2$ (figure 14 c) exhibits a number of changes under control as compared with the uncontrolled flow (figure 8 b,c). Firstly, the maximum magnitude of the $\overline{\unicode[STIX]{x1D714}_{x}^{\ast }}$ concentration associated with the C-pillar vortex is only 8, significantly lower than that (at 13) in the uncontrolled flow (figure 8 c), that is, the C-pillar vortex is weakened substantially in strength. Secondly, the streamlines (figure 14 c) show two foci, one at ( $y^{\ast }$ , $z^{\ast }$ )  $=$  ( $-0.38$ , 0.47) and the other at ( $y^{\ast }$ , $z^{\ast }$ )  $=$  ( $-0.38$ , 0.11), behind the base, which correspond to the legs of the upper and lower recirculation bubbles, respectively. Note that the values of $C_{p15}$ – $C_{p19}$ decrease under $S_{3}$ , as is the case under $S_{1}$ (figure 11 a,c). It may be inferred that the drop in $C_{p15}$ – $C_{p19}$ under $S_{3}$ is due to the proximity to the centres of the upper and lower recirculation bubbles.

At $C_{\unicode[STIX]{x1D707}}^{S3}=6.9\times 10^{-2}$ and $\unicode[STIX]{x1D703}_{S3}=-45^{\circ }$ , $S_{3}$ leads to a drag increase by approximately 20 % (figure 13 b). The time-averaged streamlines (not shown) show that the shear layer, when separated from the upper edge of the base, is deflected downwards by the blowing but, when separated from the lower edge, sweeps towards the ground without rolling up, resulting in only one recirculation bubble. This bubble expands downwards, although shrinking longitudinally, compared with the upper recirculation bubble in the base flow; meanwhile, the centre of the bubble is shifted closer to the base, accounting for the drag increase at negative $\unicode[STIX]{x1D703}_{S3}$ .

3.3.4 Actuation $S_{4}$ along the lower edge of the vertical base

With increasing $C_{\unicode[STIX]{x1D707}}^{S4}$ , $C_{D}$ decreases (figure 15 a) and $\unicode[STIX]{x0394}C_{D}$ is negatively more pronounced, down to $-14\,\%$ at $C_{\unicode[STIX]{x1D707}}^{S4}=6.9\times 10^{-2}$ for $\unicode[STIX]{x1D703}_{S4}=45^{\circ }$ (figure 15 b) but only $-2\,\%$ for $\unicode[STIX]{x1D703}_{S4}=30^{\circ }$ . At this $C_{\unicode[STIX]{x1D707}}^{S4}$ ( $\unicode[STIX]{x1D703}_{S4}=45^{\circ }$ ), $C_{p2}$ is increased by 28 % relative to the uncontrolled flow (figure 16 a). Further downstream, $C_{p3}$ , $C_{p4}$ and $C_{p5}$ rise by 14 %, 6 % and 2 %, respectively. At $y^{\ast }=0.45$ , $C_{p11}$ , $C_{p12}$ , $C_{p13}$ and $C_{p14}$ go up by 26 %, 9 %, 6 % and 6 %, respectively. The results indicate that, like $S_{3}$ , $S_{4}$ produces an effect on the flow structure over the rear window. $C_{p11}$ – $C_{p14}$ values on the base ( $y^{\ast }=0$ ) rise but $C_{p15}$ – $C_{p19}$ ( $y^{\ast }=0.45$ ) drop under $S_{4}$ . Once $\unicode[STIX]{x1D703}_{S4}\leqslant 0^{\circ }$ , $\unicode[STIX]{x0394}C_{D}$ turns positive (figure 15 b), reaching 8 % and 15 % at $C_{\unicode[STIX]{x1D707}}^{S4}=6.9\times 10^{-2}$ for $\unicode[STIX]{x1D703}_{S4}=-30^{\circ }$ and $-45^{\circ }$ , respectively.

Figure 15. Dependence on $C_{\unicode[STIX]{x1D707}}^{S4}$ of (a) $C_{D}$ and (b) $\unicode[STIX]{x0394}C_{D}$ under $S_{4}$ at $\unicode[STIX]{x1D703}_{S4}=45^{\circ }$ , $30^{\circ }$ , $0^{\circ }$ , $-30^{\circ }$ and $-45^{\circ }$ for $Re=1.7\times 10^{5}$ .

Figure 16. (a) Variation in $C_{p}$ on the rear window and the vertical base at $y^{\ast }=0$ and 0.45, and change in $\langle C_{p}\rangle$ , under $S_{4}$ ( $\unicode[STIX]{x1D703}_{S4}=45^{\circ }$ ) at $C_{\unicode[STIX]{x1D707}}^{S4}=6.9\times 10^{-2}$ , and the corresponding time-averaged flow structure (b) in the ( $x$ , $z$ ) plane of $y^{\ast }=0$ and (c) in the ( $y$ , $z$ ) plane of $x^{\ast }=0.2$ . The symbols ‘ $\times$ ’ and ‘ $+$ ’ and the red-coloured broken line in (b,c) are as in figure 11(b,c). The contour interval and cutoff levels for $\overline{\unicode[STIX]{x1D714}_{x}^{\ast }}$ in (c) are the same as in figure 11(c). $Re=1.7\times 10^{5}$ .

Several changes take place with the flow structure under $S_{4}$ operated at $C_{\unicode[STIX]{x1D707}}^{S4}=6.9\times 10^{-2}$ and $\unicode[STIX]{x1D703}_{S4}=45^{\circ }$ . Similarly to the case of $S_{3}$ at $\unicode[STIX]{x1D703}_{S3}=45^{\circ }$ , the streamlines (figure 16 b) display an enlarged separation bubble over the rear window, and longitudinally expanded upper and lower recirculation bubbles behind the base, as compared with the uncontrolled flow. One saddle point occurs above the upper edge of the base, which probably results from the interaction between the downwash flow reattaching on the slanted surface and the upwash flow near the upper edge of the base. Due to the change in the flow structure, $C_{p4}$ and $C_{p8}$ measured at the centres of the rear window and vertical base recover by approximately 6 % and 30 % (figure 16 a), respectively. Meanwhile, the strength of C-pillar vortices is substantially reduced (figure 16 c). Furthermore, the pair of foci at $y^{\ast }=\pm 0.22$ and $z^{\ast }=0.23$ shown in the streamlines (figure 8 b), which are connected to the trailing legs of the upper recirculation bubble, cannot be observed anymore under $S_{4}$ (figure 16 c), suggesting a change in the upper recirculation bubble. In fact, the streamlines in the ( $x$ , $z$ ) plane of $y^{\ast }=0.45$ (not shown) display two centres, one above the other, behind the base, which are associated with the upper and lower recirculation bubbles, respectively. The upper centre occurs at $x^{\ast }=0.33$ and $z^{\ast }=0.43$ , almost the same as that ( $x^{\ast }=0.33$ , $z^{\ast }=0.45$ ) in the symmetry plane of the wake (figure 16 b), suggesting that the legs of the upper bubble are horizontally orientated and parallel to the vertical base of the model. Due to the change in the flow structure, $C_{p15}$ – $C_{p19}$ measured near the side edge of the base decrease by 3–30 %.

The effect of $Re$ on the DR for $S_{1}$ , $S_{2}$ , $S_{3}$ and $S_{4}$ at different blowing angles is examined at $Re=1.3\times 10^{5}$ , $1.7\times 10^{5}$ and $2.0\times 10^{5}$ . Under each actuation, the dependence of $\unicode[STIX]{x0394}C_{D}$ on the momentum coefficient is mostly the same for the three different $Re$ values. This is not unexpected. The flow separation lines are fixed at the sharp edges of the rear window and the vertical base of the Ahmed body. Nevertheless, a difference in $\unicode[STIX]{x0394}C_{D}$ is appreciable under $S_{2}$ ( $\unicode[STIX]{x1D703}_{S2}=90^{\circ }$ ) as $Re$ varies. The optimal $C_{\unicode[STIX]{x1D707}}^{S2}$ is $0.7\times 10^{-2}$ at $Re=(1.7{-}2.0)\times 10^{5}$ , which is smaller than that $(1.2\times 10^{-2})$ at $Re=1.3\times 10^{5}$ (not shown). A smaller $Re$ is associated with an increase in the separation region and hence a larger low pressure area on the rear window in the absence of control (Joseph et al. Reference Joseph, Amandolese and Aider2012), which is confirmed by an increase in $C_{D}$ from 0.35 to 0.37 as $Re$ drops from $2.0\times 10^{5}$ to $1.3\times 10^{5}$ (figure 6). As a result, the pressure difference between flows on the rear window and the side surface is augmented, producing an increased strength in the C-pillar vortices. As such, a larger $C_{\unicode[STIX]{x1D707}}^{S2}$ is required to break the C-pillar vortices at $Re=1.3\times 10^{5}$ , as compared with the higher $Re$ . A difference in $\unicode[STIX]{x0394}C_{D}$ is also discernible under $S_{1}$ at $\unicode[STIX]{x1D703}_{S1}=65^{\circ }$ once $C_{\unicode[STIX]{x1D707}}^{S1}\geqslant 1.1\times 10^{-2}$ (not shown). As under $S_{2}$ ( $\unicode[STIX]{x1D703}_{S2}=90^{\circ }$ ), the optimal $C_{\unicode[STIX]{x1D707}}^{S1}$ is higher for larger $Re$ . That is, the observed $Re$ effects are mainly due to the $Re$ dependence of the flow separation bubble on the slanted surface.

3.3.5 Unsteady structures emanating from the two recirculation bubbles behind the base

The value of $E_{u}$ (figure 17 a) of $u_{xz}$ measured at location W ( $x^{\ast }=0.4$ , $y^{\ast }=0$ , $z^{\ast }=0.18$ ) displays a peak at $f^{\ast }=0.52$ without control ( $C_{\unicode[STIX]{x1D707}}^{S1}=0$ ). This peak results from the alternate emanation of structures from the upper and lower recirculation bubbles behind the base. Under the actuation of $S_{1}$ ( $\unicode[STIX]{x1D703}_{S1}=65^{\circ }$ ) this peak remains pronounced at $C_{\unicode[STIX]{x1D707}}^{S1}=0.1\times 10^{-2}$ and $0.2\times 10^{-2}$ but retreats at $C_{\unicode[STIX]{x1D707}}^{S1}=0.4\times 10^{-2}$ . The peak at $f^{\ast }=0.52$ in $E_{u}$ vanishes as $C_{\unicode[STIX]{x1D707}}^{S1}$ is further increased, implying that the alternate emanation of structures is interrupted by the control. Note that at this $C_{\unicode[STIX]{x1D707}}^{S1}$ , $C_{p8}$ at the centre of the base increases by approximately 22 % (figure 11 a) and drag reduces by 12 % (figure 10 b). This result indicates that the alternately emanated structures from the two bubbles are closely connected to the base pressure and hence the aerodynamic drag of the Ahmed body. Zhang et al. (Reference Zhang, Zhou and To2015) further found that the recirculation bubbles were characterized by periodic enlargement and contraction, and the coherent structures were alternately emanating from the two bubbles. They proposed that the bubble, be it upper or lower, expanded in size as a result of entrainment of the separated flow, which caused a gradual rise in pressure within the bubble. The bubble eventually burst when the pressure reached a certain level, accompanied by a substantially reduced size of the bubble and a structure emanated from it. At the same time, the other bubble continued its increase in size and pressure until its collapse, associated with a downsize and the emanation of a structure from it. However, under control, an increase in the size of the two recirculation bubbles implies relatively more stagnant fluid within the bubbles. As a result, the dynamic process of the bubble growing and bursting is interrupted and even suppressed, which is corroborated by the disappearance of the alternate emanation of the coherent structures from the two bubbles.

Figure 17. $E_{u}$ measured at W ( $x^{\ast }=0.4$ , $y^{\ast }=0$ , $z^{\ast }=0.18$ ) under (a) $S_{1}$ ( $\unicode[STIX]{x1D703}_{S1}=65^{\circ }$ ), (b) $S_{2}$ ( $\unicode[STIX]{x1D703}_{S2}=90^{\circ }$ ), (c) $S_{3}$ ( $\unicode[STIX]{x1D703}_{S3}=45^{\circ }$ ) and (d) $S_{4}$ ( $\unicode[STIX]{x1D703}_{S4}=45^{\circ }$ ) for different momentum coefficients. $Re=1.7\times 10^{5}$ .

The peak at $f^{\ast }=0.52$ in $E_{u}$ is discernible under $S_{2}$ at $\unicode[STIX]{x1D703}_{S2}=90^{\circ }$ , regardless of the momentum coefficient value (figure 17 b). Apparently, the actuation fails to break the organized structures. The corresponding DR is rather limited, around 6 % (figure 12 b). When $S_{3}$ is operated at $\unicode[STIX]{x1D703}_{S3}=45^{\circ }$ , the peak at $f^{\ast }=0.52$ in $E_{u}$ remains discernible up to $C_{\unicode[STIX]{x1D707}}^{S3}=0.1\times 10^{-2}$ but appears weaker at $C_{\unicode[STIX]{x1D707}}^{S3}=0.8\times 10^{-2}$ and disappears at $C_{\unicode[STIX]{x1D707}}^{S3}\geqslant 1.9\times 10^{-2}$ (figure 17 c). Similarly, under $S_{4}$ at $\unicode[STIX]{x1D703}_{S4}=45^{\circ }$ , the $E_{u}$ (figure 17 d) displays a peak at $f^{\ast }=0.52$ , which becomes weak and then vanishes from $C_{\unicode[STIX]{x1D707}}^{S4}=0.1\times 10^{-2}$ to $4.6\times 10^{-2}$ . The disappearance of the alternately emanated structures from the two recirculation bubbles under $S_{3}$ or $S_{4}$ leads to a significant reduction in drag by 10 % and 12 % at a momentum coefficient of $4.6\times 10^{-2}$ (figures 13 b and 15 b), respectively.

The effect of the blowing angle on the unsteady structure of $St=0.52$ is investigated. The results are summarized in figure 18. A number of observations can be made. Firstly, $S_{1}$ is effective in breaking up the unsteady structures of $St=0.52$ at $\unicode[STIX]{x1D703}_{S1}=90^{\circ }$ , as at $\unicode[STIX]{x1D703}_{S1}=65^{\circ }$ , given an adequately large $C_{\unicode[STIX]{x1D707}}^{S1}$ , but not at $\unicode[STIX]{x1D703}_{S1}=0^{\circ }$ and $30^{\circ }$ . Secondly, although ineffective at $\unicode[STIX]{x1D703}_{S2}=60^{\circ }$ and $90^{\circ }$ , $S_{2}$ at $\unicode[STIX]{x1D703}_{S2}=30^{\circ }$ may eradicate the structures provided $C_{\unicode[STIX]{x1D707}}^{S2}\geqslant 1.6\times 10^{-2}$ . Thirdly, $S_{3}$ at $\unicode[STIX]{x1D703}_{S3}=45^{\circ }$ is effective in eradicating the unsteady structures given $C_{\unicode[STIX]{x1D707}}^{S3}\geqslant 1.9\times 10^{-2}$ as the two bubbles behind the base are considerably enlarged (figure 14 b). So does $S_{3}$ at $\unicode[STIX]{x1D703}_{S3}=-45^{\circ }$ for very large $C_{\unicode[STIX]{x1D707}}^{S3}\;({\geqslant}3.5\times 10^{-2})$ , although the physics behind this is different, the upper bubble shrinks greatly. Finally, $S_{4}$ may effectively destroy the structures at positive $\unicode[STIX]{x1D703}_{S4}$ (figure 16 b), but not at $\unicode[STIX]{x1D703}_{S4}\leqslant 0^{\circ }$ .

Figure 18. Momentum coefficients and blowing angles of (a) $S_{1}$ , (b) $S_{2}$ , (c) $S_{3}$ and (d) $S_{4}$ , where the predominant vortex frequency $(St=0.52)$ is detected at W. Please refer to figure 17 for the location of W. The symbol ‘ $\times$ ’ denotes the occurrence of the frequency. $Re=1.7\times 10^{5}$ .

3.4.1 Control performance

As documented in detail by Zhang et al. (Reference Zhang, Zhou and To2015), the Ahmed body wake is highly complicated and characterized by distinct organized motions such as the C-pillar vortices, separation bubble over the rear window, two recirculation bubbles behind the base, alternately emanating structures from the two bubbles. Furthermore, these organized structures are coupled, that is, manipulating one may affect one or more others. Consequently, while suppressing one or two organized structures and associated drag, the actuation may alter another structure, which contributes to a rise in drag (Brunn et al. Reference Brunn, Wassen, Sperber, Nitsche and Thiele2007). Naturally, individual actuations $S_{1}$ , $S_{2}$ , $S_{3}$ and $S_{4}$ , each manipulating one or two types of organized structures, may have difficulty in control effectively all or most of the organized structures and thus achieve only rather limited DR, as demonstrated in § 3.3. One issue arises, that is, can we find a technique that combines the individual actuations to manipulate all or most of the organized structures for a significantly better control performance? In this section, we explore different combinations of individual actuations to maximize DR at $Re=1.7\times 10^{5}$ . Three schemes will be examined, i.e. combinations of $S_{1}$ and $S_{2}$ , $S_{3}$ and $S_{4}$ , and $S_{1}$ , $S_{2}$ , $S_{3}$ and $S_{4}$ . $S_{1}$ and $S_{2}$ are directed normally to the slanted surface, i.e.  $\unicode[STIX]{x1D703}_{S1}=65^{\circ }$ and $\unicode[STIX]{x1D703}_{S2}=90^{\circ }$ , and $S_{3}$ and $S_{4}$ are operated at $\unicode[STIX]{x1D703}_{S3}=\unicode[STIX]{x1D703}_{S4}=45^{\circ }$ . Such orientations, along with the fact that each actuation has its own independent air passage/chamber, lead to little interference between individual actuations.

As presented in § 3.3, $S_{1}$ may produce three effects, i.e. the downstream shift of the flow separation from the upper edge of the rear window, the longitudinal enlargement of the upper and lower recirculation bubbles, and the interruption of the alternate emanation of structures from the two bubbles, thus resulting in a DR by 12 %. On the other hand, $S_{2}$ suppresses largely only the C-pillar vortices, leading to a maximum DR of 6 %. However, there exists a coupling between the C-pillar vortices and the separation bubble over the rear window (Brunn et al. Reference Brunn, Wassen, Sperber, Nitsche and Thiele2007). In view of this, we deploy a combination of actuations $S_{1}$ and $S_{2}$ to control simultaneously the two types of structures. $C_{D}$ and $\unicode[STIX]{x0394}C_{D}$ depend on $C_{\unicode[STIX]{x1D707}}^{S1}$ and $C_{\unicode[STIX]{x1D707}}^{S2}$ , as shown in figure 19  $(Re=1.7\times 10^{5})$ . The experimental uncertainties of $C_{D}$ and its variation are estimated to be less than 0.003 and 0.007, respectively. The combined $S_{1}$ and $S_{2}$ always lead to DRs. The $C_{D}$ drops substantially compared with the uncontrolled flow. Two local maxima occur, one at ( $C_{\unicode[STIX]{x1D707}}^{S1}$ , $C_{\unicode[STIX]{x1D707}}^{S2}$ )  $=$  ( $0.3\times 10^{-2},2.0\times 10^{-2}$ ) and the other at ( $1.2\times 10^{-2}$ , $1.3\times 10^{-2}$ ), reaching 16 % and 14 % (figure 19 b), respectively. The former exceeds any previously reported DR using active techniques (table 1), where the maximum DR obtained is 14 % (Aubrun et al. Reference Aubrun, Mcnally, Alvi and Kourta2011). Evidently, the overly large $C_{\unicode[STIX]{x1D707}}^{S1}$ and $C_{\unicode[STIX]{x1D707}}^{S2}$ may lead to a deterioration in control performance. The changes in the flow structure under this combined actuation will be discussed later based on the surface pressure data.

Figure 19. Dependence on $C_{\unicode[STIX]{x1D707}}^{S1}$ and $C_{\unicode[STIX]{x1D707}}^{S2}$ of (a) $C_{D}$ and (b) $\unicode[STIX]{x0394}C_{D}$ under combined $S_{1}$ ( $\unicode[STIX]{x1D703}_{S1}=65^{\circ }$ ) and $S_{2}$ ( $\unicode[STIX]{x1D703}_{S2}=90^{\circ }$ ). (c) Variation in $C_{p}$ on the rear window and the vertical base at $y^{\ast }=0$ and 0.45, and change in $\langle C_{p}\rangle$ , with the combined actuations at $C_{\unicode[STIX]{x1D707}}^{S1}=0.3\times 10^{-2}$ and $C_{\unicode[STIX]{x1D707}}^{S2}=2.0\times 10^{-2}$ . $Re=1.7\times 10^{5}$ .

At $C_{\unicode[STIX]{x1D707}}^{S1}=0.3\times 10^{-2}$ and $C_{\unicode[STIX]{x1D707}}^{S2}=2.0\times 10^{-2}$ , $C_{p2}$ , $C_{p3}$ , $C_{p4}$ and $C_{p5}$ at $y^{\ast }=0$ are increased by approximately 39 %, 19 %, 10 % and 2 % (figure 19 c), respectively. Note that the magnitude of $\unicode[STIX]{x0394}C_{p2}$ under combined $S_{1}$ and $S_{2}$ is smaller than that (43 %) under $S_{1}$ (figure 11 a). This is reasonable since the weakened C-pillar vortices under the combined $S_{1}$ and $S_{2}$ may lead to an increased flow separation over the rear window because the two structures are coupled. $C_{p11}$ , $C_{p12}$ , $C_{p13}$ and $C_{p14}$ near the side edge ( $y^{\ast }=0.45$ ) of the slanted surface also go up by 35 %, 13 %, 10 % and 7 %, respectively. The change in the upper and lower recirculation bubbles behind the base under this combined actuation bears a similarity to that under $S_{1}$ (figure 11 b,c). The two bubbles expand longitudinally under control in the symmetry plane, which accounts for the rise in $C_{p6}$ – $C_{p10}$ on the base at $y^{\ast }=0$ . Near the side edge of the base, the legs of the two bubbles are tilted upstream toward the base due to control, leading to a decrease in $C_{p15}$ – $C_{p19}$ at $y^{\ast }=0.45$ , which are in close proximity to the centre of the two bubbles. The averaged pressure over $C_{p2}$ – $C_{p19}$ is increased by 15 % under this combination.

No matter whether $S_{3}$ or $S_{4}$ is deployed at $\unicode[STIX]{x1D703}_{S3}=\unicode[STIX]{x1D703}_{S4}=45^{\circ }$ , the substantial DR (figures 13 and 15) is associated with the suppression of the C-pillar vortices (figures 14 c and 16 c), the longitudinally enlarged recirculation bubbles behind the base (figures 14 b and 16 b) and the elimination of the alternately emanated structures from the two bubbles (figure 17 c,d). When the two actuations are combined, the $C_{D}$ reaches its minimum of approximately 0.30 at $C_{\unicode[STIX]{x1D707}}^{S3}=8.2\times 10^{-2}$ and $C_{\unicode[STIX]{x1D707}}^{S4}=6.9\times 10^{-2}$ (figure 20 a), producing a maximum DR of 18 % (figure 20 b). Correspondingly, $C_{p2}$ rises by 43 % (figure 20 c), higher than those (27 % and 28 %) produced under $S_{3}$ (figure 14 a) and $S_{4}$ (figure 16 a). Further downstream, $C_{p3}$ and $C_{p4}$ are increased by 22 % and 10 %, respectively, although $C_{p5}$ near the lower edge of the slanted surface is almost unchanged. $C_{p11}$ – $C_{p14}$ near the side edge of the rear window also rise by 7–40 %. The values of $C_{p6}$ – $C_{p10}$ and $C_{p15}$ – $C_{p19}$ increase and decrease, respectively, relatively to the uncontrolled flow. Their average rises by 17 % under control.

Figure 20. Dependence on $C_{\unicode[STIX]{x1D707}}^{S3}$ and $C_{\unicode[STIX]{x1D707}}^{S4}$ of (a) $C_{D}$ and (b) $\unicode[STIX]{x0394}C_{D}$ under combined $S_{3}$ ( $\unicode[STIX]{x1D703}_{S3}=45^{\circ }$ ) and $S_{4}$ ( $\unicode[STIX]{x1D703}_{S4}=45^{\circ }$ ). (c) Variation in $C_{p}$ on the rear window and the vertical base at $y^{\ast }=0$ and 0.45, and change in $\langle C_{p}\rangle$ , with the combined actuations at $C_{\unicode[STIX]{x1D707}}^{S3}=8.2\times 10^{-2}$ and $C_{\unicode[STIX]{x1D707}}^{S4}=6.9\times 10^{-2}$ . $Re=1.7\times 10^{5}$ .

To enhance further the DR performance, the combination of $S_{1}$ , $S_{2}$ , $S_{3}$ and $S_{4}$ is deployed to control simultaneously the separation bubble and the C-pillar vortices over the rear window and the two recirculation bubbles behind the base. Under this scheme, $C_{\unicode[STIX]{x1D707}}^{S1}$ and $C_{\unicode[STIX]{x1D707}}^{S2}$ are set at $0.3\times 10^{-2}$ and $2.0\times 10^{-2}$ , respectively, at which a DR of 16 % was obtained provided that $S_{1}$ and $S_{2}$ are simultaneously implemented. Figure 21(a,b) presents the contours of the dependence of $C_{D}$ and $\unicode[STIX]{x0394}C_{D}$ on $C_{\unicode[STIX]{x1D707}}^{S3}$ and $C_{\unicode[STIX]{x1D707}}^{S4}$ . At $C_{\unicode[STIX]{x1D707}}^{S4}=0$ , i.e. under the combination of $S_{1}$ , $S_{2}$ and $S_{3}$ , $\unicode[STIX]{x0394}C_{D}$ drops with increasing $C_{\unicode[STIX]{x1D707}}^{S3}$ and reaches the minimum, $-25\,\%$ , at $C_{\unicode[STIX]{x1D707}}^{S3}=5.6\times 10^{-2}$ , beyond which there is no further decrease in drag. On the other hand, at $C_{\unicode[STIX]{x1D707}}^{S3}=0$ or under the combination of $S_{1}$ , $S_{2}$ and $S_{4}$ , $\unicode[STIX]{x0394}C_{D}$ declines with increasing $C_{\unicode[STIX]{x1D707}}^{S4}$ , reaching the minimum, $-26\,\%$ , at $C_{\unicode[STIX]{x1D707}}^{S4}=8.2\times 10^{-2}$ , beyond which $\unicode[STIX]{x0394}C_{D}$ increases gradually for higher $C_{\unicode[STIX]{x1D707}}^{S4}$ . Finally, as $S_{1}$ , $S_{2}$ , $S_{3}$ and $S_{4}$ are all operated, a maximum reduction in drag of 29 % is obtained at $C_{\unicode[STIX]{x1D707}}^{S3}=1.9\times 10^{-2}$ and $C_{\unicode[STIX]{x1D707}}^{S4}=8.2\times 10^{-2}$ . This is higher than any previously reported DR and in fact very close to the target (30 %) set by the automotive industries (Bruneau et al. Reference Bruneau, Creuse, Delphine, Gilliéron and Mortazavi2011). Under this combination, there is a large pressure recovery about the centre of the rear window (figure 21 c), $\unicode[STIX]{x0394}C_{p2}$ , $\unicode[STIX]{x0394}C_{p3}$ , $\unicode[STIX]{x0394}C_{p4}$ and $\unicode[STIX]{x0394}C_{p5}$ reaching 55 %, 39 %, 31 % and 21 %, respectively. It is worth noting that $\unicode[STIX]{x0394}C_{p5}$ is significantly larger than those ( ${<}2\,\%$ ) under the four individual actuations and the two combined actuations ( $S_{1}$ and $S_{2}$ or $S_{3}$ and $S_{4}$ ), as shown in figures 11(a), 14(a), 16(a), 19(c) and 20(c). It will be shown later that this marked pressure recovery is associated with the entrainment of control-generated coherent structures, formed near the lower end of the slanted surface, into the separation bubble over the rear window. On the other hand, $C_{p11}$ , $C_{p12}$ , $C_{p13}$ and $C_{p14}$ near the side edge of the rear window are increased by 45 %, 27 %, 23 % and 22 %, respectively. The $C_{p6}$ – $C_{p10}$ values about the symmetry plane of the base all rise by 23–38 %, although $C_{p15}$ – $C_{p19}$ near the side edge of the base are essentially the same as those in the uncontrolled flow. Overall, the averaged pressure rises by 28 %.

Figure 21. Dependence on $C_{\unicode[STIX]{x1D707}}^{S3}$ and $C_{\unicode[STIX]{x1D707}}^{S4}$ of (a) $C_{D}$ and (b) $\unicode[STIX]{x0394}C_{D}$ under combined $S_{1}$ ( $\unicode[STIX]{x1D703}_{S1}=65^{\circ }$ ),  $S_{2}$ ( $\unicode[STIX]{x1D703}_{S2}=90^{\circ }$ ),  $S_{3}$ ( $\unicode[STIX]{x1D703}_{S3}=45^{\circ }$ ) and $S_{4}$ ( $\unicode[STIX]{x1D703}_{S4}=45^{\circ }$ ). (c) Variation in $C_{p}$ on the rear window and the vertical base at $y^{\ast }=0$ and 0.45, and change in $\langle C_{p}\rangle$ , with the combined actuations at $C_{\unicode[STIX]{x1D707}}^{S1}=0.3\times 10^{-2}$ , $C_{\unicode[STIX]{x1D707}}^{S2}=2.0\times 10^{-2}$ , $C_{\unicode[STIX]{x1D707}}^{S3}=1.9\times 10^{-2}$ , $C_{\unicode[STIX]{x1D707}}^{S4}=8.2\times 10^{-2}$ . $Re=1.7\times 10^{5}$ .

Table 2 summarizes the maximum DRs and pressure recoveries on the rear window and the vertical base under the four individual actuations $S_{1}$ , $S_{2}$ , $S_{3}$ and $S_{4}$ , and their combinations. The maximum reductions in drag produced by $S_{1}$ , $S_{2}$ , $S_{3}$ and $S_{4}$ are 12 %, 6 %, 11 % and 14 %, respectively. The combined actuations ( $S_{1}$ and $S_{2}$ ), ( $S_{3}$ and $S_{4}$ ), ( $S_{1}$ , $S_{2}$ and $S_{3}$ ) and ( $S_{1}$ , $S_{2}$ and $S_{4}$ ) produce DRs by 16 %, 18 %, 25 % and 26 %, respectively. The combined $S_{1}$ , $S_{2}$ , $S_{3}$ and $S_{4}$ achieves the highest reduction in drag.

Table 2. DR and pressure recovery achieved under individual $S_{1}$ ( $\unicode[STIX]{x1D703}_{S1}=65^{\circ }$ ),  $S_{2}$ ( $\unicode[STIX]{x1D703}_{S2}=90^{\circ }$ ),  $S_{3}$ ( $\unicode[STIX]{x1D703}_{S3}=45^{\circ }$ ) and $S_{4}$ ( $\unicode[STIX]{x1D703}_{S4}=45^{\circ }$ ) and their combinations ( $Re=1.7\times 10^{5}$ ).  $\langle C_{p}\rangle _{CR}$ and $\langle C_{p}\rangle _{LR}$ are spatially averaged pressure coefficients from $C_{p2}{-}C_{p5}$ measured at the centreline ( $y^{\ast }=0$ ) and $C_{p11}{-}C_{p14}$ near the side edge ( $y^{\ast }=0.45$ ) of the rear window, respectively, and $\langle C_{p}\rangle _{CB}$ and $\langle C_{p}\rangle _{LB}$ are the coefficients based on $C_{p6}{-}C_{p10}$ at the centreline ( $y^{\ast }=0$ ) and $C_{p15}{-}C_{p19}$ near the side edge ( $y^{\ast }=0.45$ ) of the vertical base, respectively.

3.4.2 Altered flow structure

This section is focused on the altered flow structure that corresponds to the maximum DR of 29 %, with a view to understanding the DR mechanisms. Figure 22 presents the altered flow structure in the symmetry plane under the combination of $S_{1}$ , $S_{2}$ , $S_{3}$ and $S_{4}$ , which yields the maximum DR of 29 %. The changes (figure 22 a,c) in flow separation over the rear window and the two recirculation bubbles behind the base resemble those under the individual $S_{1}$ , $S_{3}$ and $S_{4}$ cases (figures 11 b, 14 b and 16 b). Nevertheless, there are differences. Firstly, one strip of the positive $\overline{\unicode[STIX]{x1D714}_{y}^{\ast }}$ concentration is seen attaching to the rear window, with its highest magnitude reaching 5 (figure 22 b). This positive vorticity strip is linked to the impressive increase in pressure on the rear window, of 31 % in $C_{p4}$ at the centre (figure 21 c). One scenario is proposed for the observations. Under the combination, especially $S_{3}$ and $S_{4}$ , the two recirculation bubbles behind the base grow in size substantially, taking their centres downstream. The increased distance between the centres and the base acts to increase significantly the base pressure (figure 21 c). Meanwhile, the separation bubble on the rear window and the upper recirculation bubble behind the base are connected or merge into one, as supported by figure 22 (cf. figure 7 b), and then the higher base pressure than that on the window (figure 21 c) produces an upward roll up (figure 22 c,d), corresponding to the positive $\overline{\unicode[STIX]{x1D714}_{y}^{\ast }}$ , accounting for the positive vorticity strip. With the two bubbles joining together, the significantly higher base pressure pushes up the pressure on the window, as noted in figure 21(c).

Figure 22. (a,c) Time-averaged streamlines and (b,d) $\overline{\unicode[STIX]{x1D714}_{y}^{\ast }}$ -contours (the contour interval  $=$  1, the cutoff level  $=$   $\pm 2$ ) measured in the ( $x$ , $z$ ) plane of $y^{\ast }=0$ under the combination of $S_{1}$ ( $\unicode[STIX]{x1D703}_{S1}=65^{\circ }$ ),  $S_{2}$ ( $\unicode[STIX]{x1D703}_{S2}=90^{\circ }$ ),  $S_{3}$ ( $\unicode[STIX]{x1D703}_{S3}=45^{\circ }$ ) and $S_{4}$ ( $\unicode[STIX]{x1D703}_{S4}=45^{\circ }$ ).  $C_{\unicode[STIX]{x1D707}}^{S1}=0.3\times 10^{-2}$ , $C_{\unicode[STIX]{x1D707}}^{S2}=2.0\times 10^{-2}$ , $C_{\unicode[STIX]{x1D707}}^{S3}=1.9\times 10^{-2}$ and $C_{\unicode[STIX]{x1D707}}^{S4}=8.2\times 10^{-2}$ ; symbols ‘ $\times$ ’ and ‘ $+$ ’ denote the saddle points and the centres; the red-coloured broken line indicates the bubble size of the base flow, determined from the time-averaged streamlines shown in figure 7(b). $Re=1.7\times 10^{5}$ .

The flow structure change under control is examined in the ( $x$ , $z$ ) plane of $y^{\ast }=0.45$ in view of the highly 3-D flow. The $\unicode[STIX]{x1D714}_{y}^{\ast }$ -contours in the base flow show two oppositely signed $\unicode[STIX]{x1D714}_{y}^{\ast }$ concentrations, enclosed by a thick contour in figure 23(a), behind the upper edge of the base. When the control is implemented, the occurrence of these $\unicode[STIX]{x1D714}_{y}^{\ast }$ concentrations is shifted upstream and upward (figure 23 b) compared with the base flow (figure 23 a). Furthermore, as shown in the $\overline{\unicode[STIX]{x1D714}_{y}^{\ast }}$ -contours (figure 23 d), both negative and positive vorticity concentrations, coinciding spatially with the C-pillar vortices, are impaired under control; their maximum magnitudes drop to 4 and 2, respectively, considerably lower than their counterparts (7, 11) in the base flow (figure 23 c).

Figure 23. Typical instantaneous $\unicode[STIX]{x1D714}_{y}^{\ast }$ (a,b) and $\overline{\unicode[STIX]{x1D714}_{y}^{\ast }}$ -contours (c,d) measured in the $(x,z)$ plane of $y^{\ast }=0.45$ : (a,c) the base flow, (b,d) under the control of combined $S_{1}$ ( $\unicode[STIX]{x1D703}_{S1}=65^{\circ }$ ),  $S_{2}$ ( $\unicode[STIX]{x1D703}_{S2}=90^{\circ }$ ),  $S_{3}$ ( $\unicode[STIX]{x1D703}_{S3}=45^{\circ }$ ) and $S_{4}$ ( $\unicode[STIX]{x1D703}_{S4}=45^{\circ }$ ) where $C_{\unicode[STIX]{x1D707}}^{S1}=0.3\times 10^{-2}$ , $C_{\unicode[STIX]{x1D707}}^{S2}=2.0\times 10^{-2}$ , $C_{\unicode[STIX]{x1D707}}^{S3}=1.9\times 10^{-2}$ and $C_{\unicode[STIX]{x1D707}}^{S4}=8.2\times 10^{-2}$ . Contour interval  $=$  1, the cutoff level  $=$   $\pm 2$ . $Re=1.7\times 10^{5}$ .

The weakened vorticity concentrations are connected to the greatly impaired C-pillar vortices under control. As shown in the $\overline{\unicode[STIX]{x1D714}_{x}^{\ast }}$ -contours in the ( $y$ , $z$ ) plane of $x^{\ast }=0.2$ (figure 24 a), the maximum magnitude of the $\overline{\unicode[STIX]{x1D714}_{x}^{\ast }}$ concentration of the C-pillar vortices is approximately 6, less than one half of that (at 13) in the base flow (figure 8 c). Note that the C-pillar vortices are formed by shear layer rolling up around the side edge of the rear window due to the pressure difference between the flow over the slanted surface and that coming off the side face of the model. Therefore, an overall increase in the surface pressure at the rear window under control (figure 21 c) may decrease this pressure difference and hence lead to the weakened C-pillar vortices. On the other hand, the C-pillar vortex centre, identified with the maximum vorticity concentration, occurs at $y^{\ast }=0.42$ and $z^{\ast }=0.69$ , which is deflected upward and toward the symmetry plane, as compared with that ( $y^{\ast }=0.45$ and $z^{\ast }=0.6$ ) in the base flow. The movement of the C-pillar vortex under control naturally causes the side vortices separated from the side edge of the rear window to shift upstream and upward, thus resulting in the upstream and upward shift in the two oppositely signed $\unicode[STIX]{x1D714}_{y}^{\ast }$ concentrations under control (figure 23 b). Near the symmetry plane, the streamlines emanate from the centre of the vertical base to the upper and lower edges of the base, which are associated with the upper and lower recirculation bubbles, respectively; please refer to the time-averaged streamlines in the ( $x$ , $z$ ) plane (figure 22 c). Nevertheless, the two recirculation bubbles are hardly discernible from the streamlines in the ( $x$ , $z$ ) plane of $y^{\ast }=0.45$ (figure 24 b). The streamlines exhibit one node near $x^{\ast }=0.15$ and $z^{\ast }=0.4$ . The flow about the node is highly three-dimensional, with a velocity gradient in the $y$ direction, as indicated by the streamlines in the ( $y$ , $z$ ) plane of $x^{\ast }=0.2$ (figure 24 a). The presence of the node is ascribed to the shear layers that sweep across this plane, indicating a transportation of fluid from the central region to the lateral side behind the base, which accounts for the 2 % rise in the spatially averaged pressure coefficient of $C_{p15}$ – $C_{p19}$ near the side edge of the base (figure 21 c). The observation from the PIV data is also substantiated by flow visualization data in the ( $y$ , $z$ ) plane at $x^{\ast }=1.4$ . One pair of counter-rotating trailing vortices is evident in the base flow (figure 24 c). The C-pillar vortex and the upper recirculation bubble interact with each other, and merge downstream, forming one pair of trailing vortices in the wake (Zhang et al. Reference Zhang, Zhou and To2015). Under control, the trailing vortices cannot be observed anymore (figure 24 d), implying greatly impaired or even eliminated vortices.

Figure 24. (a) Time-averaged streamlines and $\overline{\unicode[STIX]{x1D714}_{x}^{\ast }}$ -contours in the ( $y$ , $z$ ) plane of $x^{\ast }=0.2$ (contour interval  $=$  1, cutoff level  $=$   $\pm 1$ ), (b) time-averaged streamlines in the ( $x$ , $z$ ) plane of $y^{\ast }=0.45$ under the combination of $S_{1}$ ( $\unicode[STIX]{x1D703}_{S1}=65^{\circ }$ ),  $S_{2}$ ( $\unicode[STIX]{x1D703}_{S2}=90^{\circ }$ ),  $S_{3}$ ( $\unicode[STIX]{x1D703}_{S3}=45^{\circ }$ ) and $S_{4}$ ( $\unicode[STIX]{x1D703}_{S4}=45^{\circ }$ ). Typical photographs of the flow structure in the ( $y$ , $z$ ) plane of $x^{\ast }=1.4$ : (c) the base flow, (d) under control. The white line is the laser light reflection from the floor. $C_{\unicode[STIX]{x1D707}}^{S1}=0.3\times 10^{-2}$ , $C_{\unicode[STIX]{x1D707}}^{S2}=2.0\times 10^{-2}$ , $C_{\unicode[STIX]{x1D707}}^{S3}=1.9\times 10^{-2}$ and $C_{\unicode[STIX]{x1D707}}^{S4}=8.2\times 10^{-2}$ , $Re=1.7\times 10^{5}$ .

A conceptual model is proposed for the altered flow structure under the control of combined $S_{1}$ , $S_{2}$ , $S_{3}$ and $S_{4}$ , which leads to a great reduction in drag, as sketched in figure 25. Compared with the uncontrolled flow, the flow separation point is shifted downstream, instead of being fixed at the upper edge of the rear window (figure 22 a). The separated shear layer reattaches near the lower end of the rear window, forming a separation bubble. The downwash reattaching flow interacts with the upwash caused by blowing along the upper edge of the vertical base, producing a structure near the lower end of the slanted surface. This structure is then entrained into the separation bubble under the effect of recirculating flow over the rear window, which is responsible for the strip of positive vorticity concentrations attaching to the slanted surface (figure 22 b). Similar observations have also been made for the combination of $S_{1}$ , $S_{2}$ and $S_{3}$ . In contrast, such a positive vorticity strip cannot be observed under $S_{3}$ or $S_{4}$ , nor under their combination, in spite of the presence of a separation bubble over the rear window (figures 14 b and 16 b). Accordingly, $\unicode[STIX]{x0394}C_{p4}$ at the centre of the slanted surface reach only 4 %, 6 % and 10 % under $S_{3}$ , $S_{4}$ and ( $S_{3}+S_{4}$ ), respectively, far less than those (25 % and 31 %) produced by the combinations of ( $S_{1}$ , $S_{2}$ , and $S_{3}$ ) and ( $S_{1}$ , $S_{2}$ , $S_{3}$ and $S_{4}$ ). It is therefore plausible that the entrainment of the control-generated structure into the separation bubble over the rear window may play a significant role for the pressure rise on the slanted surface. The C-pillar vortices are significantly weakened in strength under control (figure 24 a) as compared with those in the base flow, leading to a large increase, by 23 %, in $C_{p13}$ measured near the side edge of the rear window (figure 21 c). The side vortices, formed over the side surface of the body, are wrapped up around the side edge of the rear window under the roll-up effect of the shear layer coming off the side surface along the slanted side edge (Zhang et al. Reference Zhang, Zhou and To2015). This explains why there exist two oppositely signed $\unicode[STIX]{x1D714}_{y}^{\ast }$ concentrations, coinciding spatially with the C-pillar vortices, in the typical instantaneous $\unicode[STIX]{x1D714}_{y}^{\ast }$ -contours measured in the ( $x$ , $z$ ) plane at $y^{\ast }=0.45$ (figure 23 b).

Figure 25. Conceptual model of the flow structure under the combined actuation of $S_{1}$ , $S_{2}$ , $S_{3}$ and $S_{4}$ .

The streamlines (figure 22 c) demonstrate that the upper and lower recirculation bubbles behind the base are expanded by more than 50 % (cf. figure 7 b), which is linked to the upward deflection of the separated shear layer from the upper edge of the base under $S_{3}$ . This expansion is associated with a shift both downstream and upward in the centre of the upper bubble and a downstream shift in the centre of the lower bubble; meanwhile, the saddle point moves downstream. Furthermore, the alternate emanation of the coherent structures from the two bubbles disappears. Correspondingly, $C_{p8}$ at the centre of the vertical base recovers by 38 % (figure 21 c). One strip of positive vorticity concentrations attaches to the base, as is evidenced in $\overline{\unicode[STIX]{x1D714}_{y}^{\ast }}$ -contours (figure 22 d). In the uncontrolled flow, the C-pillar vortices and the upper recirculation bubble interact with each other, forming one pair of counter-rotating trailing vortices in the wake (figure 24 c). However, the pair of trailing vortices disappear under control, as shown in the flow visualization data in the ( $y$ , $z$ ) plane of $x^{\ast }=1.4$ (figure 24 d). This is not unexpected in view of the weakened C-pillar vortices and the changes in the upper recirculation bubble behind the base.