3.1. Interaction of a synthetic jet with the boundary layer

This investigation begins by examining the PIV measurements of a synthetic jet issuing into the boundary layer of the BTE-profiled body. Although the primary region of interest in this study is the wake, the von Kármán vortex street develops from the separated shear layers, which themselves can be envisioned as continuations of the boundary layers developed over the body. Because the synthetic jets issue into the boundary layers and force the wake via the boundary layers, it is important to understand the jet–boundary layer interaction to interpret the effects on the vortex shedding.

Figure 4.

Streamwise velocity field with ( $v,w$ ) vectors in the cross-plane at $x=-1.55d$ , averaged at constant phases of the synthetic jet oscillation cycle at $Re_d=2500$ : (a) $\phi _j=258^\circ , \; R=2.7$ , (b) $\phi _j=258^\circ , \; R=3.1$ , (c) $\phi _j=258^\circ , \; R=3.9$ , (d) $\phi _j=78^\circ , \; R=2.7$ , (e) $\phi _j=78^\circ , \; R=3.1$ and (f) $\phi _j=78^\circ , \; R=3.9$ .

The blowing of a jet in a cross-flow results in a momentum exchange between the jet and the surrounding fluid and generates a wide variety of coherent vortical structures (Fric & Roshko Reference Fric and Roshko1994). Commonly, the downstream flow of a jet in cross-flow is dominated by a counter-rotating vortex pair (Karagozian Reference Karagozian2014). However, the jet trajectory and the particular vortical structures formed are strongly dependent on the relative strength of the jet as well as the jet geometry/orientation (Tricouros, Amitay & Van Buren Reference Tricouros, Amitay and Van Buren2023). For a pulsed or synthetic jet, the frequency also significantly impacts the types of vortical structures (Eroglu & Breidenthal Reference Eroglu and Breidenthal2001; Jabbal & Zhong Reference Jabbal and Zhong2008; Sau & Mahesh Reference Sau and Mahesh2008). The focus of the present study is not to map this wide parameter space, but to investigate specific forcing amplitudes that are important for distributed forcing with the current jet geometry. Because the jet geometry is fixed, the interpretation of the results in the present study is not affected by whether $R$ or $C_\mu$ is used to characterise the forcing amplitude. However, for this study, $R$ is preferred over $C_\mu$ because $R$ characterises only the strength of the forcing at a particular spanwise position along the jet, whereas $C_\mu$ incorporates both the relative momentum flux at a particular position and the area that the jet acts over. Hence, characterising the forcing amplitude with $C_\mu$ makes an assumption that $\ell$ and $\mathcal {D}$ are equally important, which cannot be tested from the present results. The focus of this section is on velocity ratios of $R=\{2.7, \; 3.1 \; \mathrm {and} \; 3.9\}$ at $Re_d=2500$ .

Figures 4 and 5 present the streamwise velocity and spanwise vorticity fields, respectively, in the cross-plane at $x=-1.55d$ ( $0.125d$ downstream of a jet slot) averaged at constant phases with respect to the synthetic jet oscillation cycle for the selected forcing amplitude cases. Figure 4(a–c) shows the field near the suction peak at $\phi _j=258^\circ$ and figure 4(d–f) that averaged near the blowing peak at $\phi _j=78^\circ$ . The measurements in this near-field region of the jet show a clear cyclic variation of the velocity at the forcing frequency. In particular, the $\overline {v}$ component in the centre region of the slot near the wall ( $|z| \leqslant 0.3d,y \leqslant 0.7d$ ) is directed towards the wall during the suction part of the cycle and vice versa during the blowing part of the cycle. Additionally, there is a significant spanwise gradient of $\overline {v}$ near the edges of the slot from $0.3d \leqslant |z| \leqslant 0.6d$ at all phases of the cycle. This is indicative of the development of edge streamwise vortices, seen directly in figure 5, which is a characteristic feature of rectangular synthetic jets that issue normal to the wall (Van Buren et al. Reference Van Buren, Beyar, Leong and Amitay2016). The sense of rotation of the streamwise vortices is such that fluid is lifted away from the wall in the jet centreline region and brought towards the wall in the regions near the slot edges. These streamwise vortices may be expected to convect downstream and to redistribute the momentum in the boundary layer, energising the regions close to the jet slot edges with higher-velocity fluid from the outer part of the boundary layers and vice versa in regions more closely aligned with the jet centreline. If the changes in the flow field between phases (time) at $x=-1.55d$ are primarily driven by the downstream convection of coherent structures through this plane, then the phase dependence could be converted into spatial information through Taylor’s hypothesis. However, the streamwise vorticity in this plane does not appear to significantly change during the cycle in spite of the significant phase variation of $\overline {v}$ near the jet edges because the average gradients in $y$ and $z$ remain relatively constant during the cycle. This implies that Taylor’s hypothesis cannot be applied in this plane. During the suction phase of all the investigated cases, $\overline {u}$ points upstream near the wall in the $x=1.55d$ plane, which is consistent with the jet ingesting the boundary layer fluid at this phase of the cycle. However, the region of reversed flow is notably larger in the $R=3.9$ case compared to the lower- $R$ cases at $Re_d=2500$ . Indeed, there is negative $\overline {u}$ in the $x=1.55d$ plane even during the blowing part of the cycle. This is indicative of the jet in the $R=3.9$ case having sufficient strength to block the incoming boundary layer and highlights a change in the jet–boundary layer interaction.

Figure 5.

Spanwise vorticity field with ( $v,w$ ) vectors in the cross-plane at $x=-1.55d$ , averaged at constant phases of the synthetic jet oscillation cycle at $Re_d=2500$ : (a) $\phi _j=258^\circ , \; R=2.7$ , (b) $\phi _j=258^\circ , \; R=3.1$ , (c) $\phi _j=258^\circ , \; R=3.9$ , (d) $\phi _j=78^\circ , \; R=2.7$ , (e) $\phi _j=78^\circ , \; R=3.1$ and (f) $\phi _j=78^\circ , \; R=3.9$ .

Figure 6.

Spanwise vorticity field with ( $u,v$ ) vectors in the streamwise plane at $z=0.05d$ , averaged at constant phases of the synthetic jet oscillation cycle at $Re_d=2500$ : (a) no forcing, (b) $\phi _j=258^\circ , \; R=2.7$ , (c) $\phi _j=258^\circ , \; R=3.1$ , (d) $\phi _j=258^\circ , \; R=3.9$ , (e) $\phi _j=78^\circ , \; R=2.7$ , (f) $\phi _j=78^\circ , \; R=3.1$ and (g) $\phi _j=78^\circ , R=3.9$ .

Figure 7.

Sketch of the dominant spanwise vorticity components from the boundary layer and a synthetic jet (during the blowing phase). (a) The vorticity from the boundary layer and synthetic jet is shown separately. (b) The synthetic jet is shown issuing into the boundary layer, which causes the vortex on the upstream side of the jet either to be cancelled or to get weaker and the vortex on the downstream side to get stronger.

The interaction of the jet and the boundary layer and its propagation downstream can be better seen in the streamwise plane over a jet slot. Figure 6 shows the velocity/vorticity fields without forcing and for the selected forcing cases in the streamwise plane at $z=0.05d$ . The forcing results are again presented near the suction peak at $\phi _j=258^\circ$ for figure 6(b–d) and near the blowing peak at $\phi _j=78^\circ$ for figure 6(e–g). Without forcing (figure 6 a), the boundary layer thickness is approximately $0.3d$ at $Re_d=2500$ ; the boundary layer shear manifests as significant negative spanwise vorticity on the upper side of the body up to $y \approx 0.75d$ . For a rectangular synthetic jet issuing in quiescent conditions, a vortex ring would be expected to be generated during the blowing phase, producing opposite-sign vorticity components at the opposite sides of the slot (Van Buren, Whalen & Amitay Reference Van Buren, Whalen and Amitay2014). This is sketched in figure 7(a). When a cross-flow boundary layer is introduced, the shear from the boundary layer opposes the positive-sign jet vorticity that would be generated on the upstream side of the slot in quiescent conditions. In the $R=2.7$ and $R=3.1$ cases at $Re_d=2500$ , the boundary layer is still largely dominated by negative spanwise vorticity, indicating that the jet vorticity on the upstream side is mostly cancelled except for some localised regions just upstream of the slot during the blowing phase, as sketched in figure 7(b). In the $R=3.9$ case, a much larger region of $+\overline {\omega }_z$ can be observed on the upstream side during the blowing phase that extends up to $y \approx 0.75d$ , indicating that the jet is sufficiently strong to overcome the boundary layer vorticity. This observation is similar to that of Jabbal & Zhong (Reference Jabbal and Zhong2008) for a synthetic jet emanating from a circular orifice and that of Sau & Mahesh (Reference Sau and Mahesh2008) for a pulsed circular jet. In particular, Jabbal & Zhong (Reference Jabbal and Zhong2008) found that the structures formed by their synthetic jet changed from hairpin vortices to distorted vortex rings when $R$ reached a critical amplitude. The distorted vortex rings generated by Jabbal & Zhong (Reference Jabbal and Zhong2008) penetrated much more deeply through the boundary layer than the hairpin vortices, which can be attributed to the inherent self-induced velocity pointing away from the wall of vortex rings. In the present study, coherent vortex rings are not evident from the synthetic jet slot in the cross-flow, consistent with the observations of Van Buren et al. (Reference Van Buren, Beyar, Leong and Amitay2016) for rectangular synthetic jets. Van Buren et al. (Reference Van Buren, Beyar, Leong and Amitay2016) attributed the absence of vortex rings to them breaking up almost immediately after they form. However, the considerable phase-coherent fluctuations at the forcing frequency in the near field of the jet are indicative of some type of coherent structure. Theoretically, if the phase-coherent fluctuations are carried by vortices of $\mathcal {O} (\mathcal {D})$ , this would correspond to only about three vectors at the current resolution. The fact that these small-scale structures are not directly evident in the PIV flow fields suggests that they are simply too small and closely packed to be resolved from the current measurements. It can be seen in figure 6 that the emergence of positive vorticity at the upstream side of the slot is associated with a considerable increase in the jet penetration compared with the lower- $R$ cases. This suggests that the emergence of a significant concentration of positive vorticity at the upstream side of the slot has a similar effect to the self-induced velocity from an actual vortex ring.

In all the forcing cases, the jets cause a streamwise velocity deficit downstream of the slots. This can be attributed to at least two different fundamental mechanisms. Firstly, the jets have a blockage-like effect because the fluid coming out from the plane of the slot is initially pointed in the wall-normal direction. Therefore, the jet transfers its wall-normal momentum to the cross-flow. By the same token, the cross-flow transfers some streamwise momentum to the jet fluid, causing the fluid downstream of the slot to slow down and the jet bends in the direction of the cross-flow until a new equilibrium is reached. Secondly, the streamwise vortices generated at the edges of the synthetic jet slots (see figure 5) are rotating in a direction to cause the low-momentum fluid aligned with the jet centreline plane to move away from the wall (Rathnasingham & Breuer Reference Rathnasingham and Breuer2003). The wake-like region downstream of the slot in the $z=0.05d$ streamwise plane is more evident for the $R=3.1$ case compared with $R=2.7$ when $Re_d=2500$ . This is consistent with the jets in the $R=3.1$ case penetrating deeper into the cross-flow and convecting downstream closer to the jet centreline plane, similar to the flow fields presented by Van Buren et al. (Reference Van Buren, Beyar, Leong and Amitay2016) for a rectangular spanwise-oriented synthetic jet. The $R=3.9$ case has a significantly larger wake-like region downstream of the slot than the lower- $R$ cases at this $Re_d$ because the jet is sufficiently strong to cause the incoming boundary layer to completely separate. Consequently, the shear layer along the jet centreline plane splits into two parts: an inner region associated with the boundary layer as it recovers downstream, and an outer region associated with the penetrating jet above which the flow finally reaches the free stream. Notably, the phase coherence of the fluctuations at the jet frequency disappears downstream of $x \approx -0.5d$ , or $\approx 35\mathcal {D}$ downstream of the slot. The loss of phase-coherent fluctuations as a synthetic jet evolves downstream has been used as an indication of the transition between the near field and far field of the jet–cross-flow interaction by Van Buren et al. (Reference Van Buren, Beyar, Leong and Amitay2016), where they estimated this transition happened at approximately $20\mathcal {D}$ downstream of their jets. In the present case, the loss of phase-coherent high-frequency jet fluctuations upstream of the trailing edge further highlights that the perturbations to the wake from the synthetic jets can justifiably be considered as steady with respect to the energy-containing structures of the wake. Importantly, these results indicate that the forcing from the array of synthetic jets is capable of significantly perturbing the boundary layers.

Figure 8.

Mean streamwise vorticity field with ( $v,w$ ) vectors in the cross-plane at $x=1d$ with (a) { $Re_d=2500$ , no forcing}, (b) $\{Re_d=2500, \; R=2.7\}$ , (c) $\{Re_d=2500, \; R=3.1\}$ , (d) $\{Re_d=2500, \; R=3.9\}$ , (e) $\{Re_d=5000, \; R=1.9\}$ and (f) $\{Re_d=5000, \; R=2.5\}$ .

3.2. Spanwise variation in the wake structure

The cross-plane in the near wake is studied next in order to observe the spanwise variations in the wake upstream of vortex street formation that are induced by the distributed forcing. figure 8 shows the distributions of the mean streamwise vorticity, $\overline {\omega }_x$ , at $x=1d$ for the unforced case at $Re_d=2500$ , as well as with forcing at $\{Re_d=2500; \; R=2.7, \; 3.1 \; \mathrm {and} \; 3.9\}$ and $\{Re_d=5000; \; R=1.9 \; \mathrm {and} \; 2.5\}$ . At this location, the shear layers are rolling up and so it is at approximately the end of what one could call the separated shear layer region. Without forcing, $\overline {\omega }_x$ is very low in this plane because the time-averaged wake is approximately spanwise uniform. Nevertheless, it should be appreciated that the wake is turbulent and exhibits instantaneous three-dimensional features, such as small-scale coherent streamwise vortices (Gibeau, Koch & Ghaemi Reference Gibeau, Koch and Ghaemi2018), but they are not observed in figure 8(a) because of the time-averaging.

Coherent streamwise vortices are evident in the cross-plane mean field of the forced cases. On each side of the body, there are two pairs of streamwise vortices in the $x=1d$ plane: one set that is biased towards the free-stream side of the separated shear layers at $|y| \gt 0.5$ , and another set at $|y| \lt 0.4d$ inside the recirculation region. In accordance with the matched forcing amplitude from all the jets and symmetric placement of the actuators on both sides of the body, the perturbations in the mean fields exhibit vertical symmetry about $y=0$ and spanwise symmetry about $z=0$ . The vortex pairs that are located at $|y| \gt 0.5$ are consistent with the jet edge streamwise vortices shown in figure 5. The sense of rotation of the vortices is such that outboard of the vortices, the higher velocity flow from the free stream is brought lower into the boundary layer, and vice versa for the region inboard of the vortices closer to the jet centreline ( $z=0$ ). Hence, it can be surmised that these vortices redistribute momentum in the boundary layer and near wake as they convect downstream.

Figure 9.

The $\partial \overline {u}/\partial y$ field with ( $v,w$ ) vectors in the cross-plane at $x=1d$ when $Re_d=2500$ with (a) no forcing, (b) $R=2.7$ , (c) $R=3.1$ and (d) $R=3.9$ .

The jet edge vortices of the $\{Re_d=2500, \; R=2.7\}$ and $\{Re_d=5000, \; R=1.9\}$ cases are located close to the jet slot edge plane. They are centred at $|y| \approx 0.5d{-}0.55d$ but extend up to $|y| \approx 0.75d$ in the $\{Re_d=2500, \; R=2.7\}$ case and to $|y| \approx 0.6d$ in the $\{Re_d=5000, \; R=1.9\}$ case. The boundary layer thickness is approximately $0.3d$ and $0.21d$ , respectively, at $Re_d=2500$ and $5000$ , so these vortices are approaching the edge of the boundary layer but are not penetrating beyond. In the $\{Re_d=2500, \; R=3.1\}$ case, the edge streamwise vortices on the $|y| \gt 0.5$ side are still mostly concentrated at $|y| \lt 0.8$ , but there are small amounts of streamwise vorticity outside of $|y| \gt 0.8$ in the jet centreline region, indicating that the streamwise vortices are penetrating right up to the edges of the boundary layers. In the $\{Re_d=2500, \; R=3.9\}$ and $\{Re_d=5000, \; R=2.5\}$ cases, the streamwise vortices formed by the jet–boundary layer interaction are larger and can be seen to be extending outside the boundary layers. For these relatively higher-forcing-amplitude cases, the pairs of vortices are more closely spaced, consistent with the findings of Van Buren et al. (Reference Van Buren, Beyar, Leong and Amitay2016) for spanwise-oriented synthetic jet slots. The fact that this occurs at a much lower $R$ when $Re_d=5000$ compared with $Re_d=2500$ suggests that $R$ (or $C_\mu$ ) is not a very appropriate way to scale the forcing amplitude across different $Re_d$ for this type of forcing. An alternative way of scaling the forcing amplitude that provides a better collapse over different $Re_d$ is proposed in § 3.3.

Figure 8 shows that on the free-stream sides of the separated shear layers ( $|y| \gt 0.5$ ), the jet edge vortices direct the flow along $z$ towards the jet centreline plane. Inside the recirculation region ( $| y | \lt 0.4d$ ), the flow points in the opposite direction, away from the jet centreline plane. This forms a set of induced streamwise vortices which have the opposite sense of rotation to that of the jet edge vortices. Although $\omega _z$ cannot be directly calculated from a cross-plane, the $\partial \overline {u}/\partial y$ field should be representative of the trends of the $\overline {\omega }_z = (\partial \overline {v}/\partial x - \partial \overline {u}/\partial y)$ field at $x=1d$ because it is still relatively close to the trailing edge. Figure 9 plots $\partial \overline {u}/\partial y$ in the $x=1d$ cross-plane for the $\{Re_d=2500; \; R = 0, \; 2.7, \; 3.1 \; \mathrm {and} \; 3.9\}$ cases. When the forcing amplitude is at $R=3.1$ , the greater penetration of the jets as compared with $R=2.7$ causes the shear layer thickness to increase more in the inboard region of the wake, as can also be seen in figure 6. The $R=3.9$ case can be contrasted to the lower- $R$ cases at $Re_d=2500$ by the presence of two distinct regions in the shear layers in the inboard region of the slots. This is consistent with the vorticity field of figure 6(d,g), and can be attributed to the boundary layers completely separating when they encounter the jet slots when $R=3.9$ at $Re_d=2500$ , thereby allowing the jets to penetrate outside of the boundary layers. This increases the width of the wake inboard of the slots and can be expected to result in a higher drag compared with the $R=2.7$ and $R=3.1$ cases at $Re_d=2500$ because the momentum deficit grows when the wake is wider. Therefore, it can be surmised that the forcing amplitude that achieves the best drag reduction should be a compromise between increasing the spanwise variations in the wake without causing it to significantly widen anywhere along the span.

Figure 10 shows profiles of $\overline {u}/u_\infty$ and $\overline {u^\prime u^\prime }/u_\infty ^2$ in the upper separated shear layer at various locations along the span for the $\{Re_d=2500, \; R = 3.1\}$ case at $x=0.2d$ , where the approximation that $-\partial \overline {u}/\partial y$ is representative of $\overline {\omega }_z$ should be more valid. It can be seen that the shear layer varies significantly along the span in the forced cases. Generally, the velocity gradient along $y$ at $y=0.5d$ is less steep in the $|z| \leqslant 0.35d$ profiles and more concentrated from $|z| \geqslant 0.45d$ compared with the unforced case. For this case, the shear layer is least concentrated, i.e. at its thickest, from $0.25d \leqslant z \leqslant 0.35d$ . The spanwise variations in the shear layer thickness are driven by the combined effect of the jet blockage and the regions of upwash and downwash associated with the streamwise vortices that redistribute the momentum in the boundary layers. The influence of the jet edge vortices can be clearly seen in the $z=0.45d$ and $0.55d$ shear layer profiles of the $\{Re_d=2500, \; R = 3.1\}$ case, where there is a velocity deficit in the outer region of the shear layer and accelerated velocity closer to $y=0.5d$ . The profile at $z=0.8d$ also exhibits a steeper velocity gradient along $y$ , but the effect diminishes moving away from the slot towards $z=1.1d$ .

Figure 10.

Selected shear layer profiles along the span at $x = 0.2d$ for $\{Re_d=2500, \; R = 3.1\}$ , measured using HWA: (a) $\overline {u}/u_\infty$ ; (b) $\overline {u^\prime u^\prime }/u_\infty ^2$ .

Given that vortex shedding is periodic, further insight into the origin of the streamwise vortices in the $x=1d$ cross-plane can be gained by phase-averaging the measurements. Using the methodology outlined in Appendix A, the phase reference for this plane is based on the relationship between the $u$ and $v$ components in the instantaneous snapshots at $\{x,y\}=\{1d,\pm 1.05d\}$ , averaged across the spanwise domain of the snapshots. This methodology is possible because the turbulent velocity fluctuations are not very large in this part of the wake and the $u$ and $v$ components are periodically modulated by the passage of the von Kármán vortices. The snapshots in the $x=1d$ plane for the $\{Re_d=2500, \; R = 3.1\}$ case were sorted with respect to the phase reference into 12 phase bins. For a given shedding phase bin, $\phi _v$ , composed of $N_\phi$ samples, the phase-averaged velocity is calculated as

(3.1) \begin{equation} \tilde {u}(\phi _v) = \sum _{i=1}^{N_\phi } \frac {u_i(\phi _v)}{N_\phi }, \end{equation}

adopting a notation where phase-averaged statistics are denoted with a tilde. The streamwise vorticity fields of the $\{Re_d=2500, \; R = 3.1\}$ case at two different shedding phases $180^\circ$ apart are shown in figure 11. Throughout the shedding cycle, the streamwise vortex pairs at $|y|\gt 0.5d$ are relatively stable in strength and location in this plane. Their consistency can be attributed to the synthetic jets operating at a much higher frequency than the vortex shedding. Hence, the steadiness of these vortices over the shedding cycle is another indication that the outer-region vortex pairs originate from the jet–boundary layer interaction and convect into the wake. In contrast, the streamwise vortices in the recirculation region periodically alternate between the upper and lower sides of the wake during the cycle, indicating a much closer connection to the wake dynamics. The phases presented in figures 11(a) and 11(b) correspond to when these vortices are at maximum strength in the upper and lower sides of the wake, respectively.

Figure 11.

Streamwise vorticity fields and $(v,w)$ vectors for the $\{Re_d=2500, \; R=3.1\}$ case in the $x=1d$ plane at (a,b) two different phases ( $180^\circ$ apart) of the shedding cycle.

Figure 12.

Sketch of the vortical structures induced by forcing in the separated shear layer from the $y\gt 0$ side of the body.

Figure 12 presents a sketch of the mean flow field in the near wake, showing the spanwise variation of the shear layer thickness and coherent vortices based on figures 8 to 11. The process by which the spanwise variations in the shear layers affect the vortex shedding and periodically induce streamwise vortices to form can be illustrated by considering the vorticity transport equation for incompressible flow, neglecting viscous diffusion:

(3.2) \begin{equation} \frac {D \omega _i}{D t} = \omega _\kappa \frac {\partial u_i}{\partial x_\kappa }, \end{equation}

and for the streamwise component, this can be decomposed as

(3.3) \begin{equation} \frac {D \omega _x}{D t} = \omega _x \frac {\partial u}{\partial x} + \omega _y \frac {\partial u}{\partial y} + \omega _z \frac {\partial u}{\partial z}. \end{equation}

Special attention is drawn to the $\omega _z (\partial u/\partial z)$ vortex tilting term because of the inherently large $\omega _z$ concentrations which compose the separated shear layers and from which the von Kármán vortices develop. The establishment of spanwise gradients in the streamwise velocity inside the separated shear layers provides the condition necessary to tilt the spanwise von Kármán vortices into the streamwise direction and to cause gradients in the spanwise vorticity along the span. It is noted that conservation of circulation requires the generation of streamwise vorticity through the diversion of spanwise vorticity to be associated with a corresponding reduction in the overall spanwise circulation. The process of shear layer tilting on the upper side of the body is sketched in figure 13. The region of $\partial u/\partial z \lt 0$ in the separated shear layer on the $z\lt 0$ side of the jet creates $+\omega _x$ , and vice versa for the other side. Hence, the direction of the tilting of the spanwise vorticity is such that the von Kármán vortices are retarded in the inboard region of the jet slots as they shed. The sense of rotation of the streamwise vortices within the recirculation region in figures 8 and 11 and that are sketched in figure 12 are consistent with this vortex street rollup pattern. As a von Kármán vortex from one side of the body is developed, the streamwise vortex pair that is developed due to the tilting of the vorticity of the separated shear layer also grows. Then, the streamwise vorticity declines on that side of the body after the von Kármán vortex sheds while the reverse of this process occurs in the shear layer from the opposite of the body. The resulting structure of the wake downstream of the vortex formation region is examined in § 3.3.

Figure 13.

Sketch of the tilting of the spanwise vorticity shed from the upper side of the body due to spanwise gradients in the streamwise velocity in the shear layer, from a top view.

Power spectral density of the velocity is used to measure the effect of forcing on the spectral characteristics and structure of the wake. The power spectra at $\{x,y\}=\{4.5d,0.8d\}$ for $\{Re_d=2500; \; R=0, \; 3.1 \; \mathrm {and} \; 3.9\}$ are presented in figures 14(a) and 14(b), respectively. Without forcing, the power spectrum is dominated by a vortex shedding frequency peak at $f_vd/u_\infty = \widehat {\textit {St}}_v=0.25$ . This value of $\widehat {\textit {St}}_v$ is comparable with that measured by Bearman (Reference Bearman1965) and Petrusma & Gai (Reference Petrusma and Gai1996) for similar BTE body geometries. Clearly the vortex shedding is attenuated in the forcing cases, but also importantly, $\widehat {\textit {St}}_v$ is constant along the span. Interestingly, the power at most frequencies other than $\widehat {\textit {St}}_v$ is increased by forcing in the $R=3.9$ case. This implies that the break-up of the vortex street in that case also generated a series of smaller-scale structures in the wake, which cascade to even smaller scales. The main difference that is evident from the power spectra between the $R=3.1$ and $R=3.9$ cases at $Re_d=2500$ is that the shedding frequency increases to $\widehat {\textit {St}}_v \approx 0.26$ for $R=3.9$ . Additionally, it is found that further increases to $R$ drive even higher $\widehat {\textit {St}}_v$ and it does not appear to plateau at least up to $R=5$ , although these results are not presented in this paper for conciseness. An increase in $\widehat {\textit {St}}_v$ is considered undesirable for drag reduction in the present context because it implies that the rate of circulation entering the wake rises.

Figure 14.

Power spectral density of the velocity, measured along the span at $\{x=4.5d,\; y=-0.8d\}$ at $Re_d=2500$ for (a) $R=3.1$ and (b) $R=3.9$ . Here $z=0$ corresponds to the jet-centreline plane.

A spanwise constant $\widehat {\textit {St}}_v$ has also been reported in some other distributed forcing studies with different types of spanwise perturbations, such as Park et al. (Reference Park, Lee, Jeon, Hahn, Kim, Kim, Choi and Choi2006) with a series of tabs on a BTE-profiled body, Dobre et al. (Reference Dobre, Hangan and Vickery2006) for a wavy upstream face on a square cylinder and Lam et al. (Reference Lam, Wang, Li and So2004) for a wavy cylinder. However, many other distributed forcing studies including Tombazis & Bearman (Reference Tombazis and Bearman1997), Bearman & Owen (Reference Bearman and Owen1998), Bhattacharya & Gregory (Reference Bhattacharya and Gregory2018) and Ling & Zhao (Reference Ling and Zhao2009) have measured two $\widehat {\textit {St}}_v$ . Spanwise $\widehat {\textit {St}}_v$ variations cause the phase difference between two locations in the wake to change over time. When the phase difference is small, the vortices can bend along the span but when the difference approaches $180 ^\circ$ , the vortex shedding desynchronises along the span and the vortices break. This behaviour was referred to as vortex dislocations by Williamson (Reference Williamson1992), and results in a redistribution of the vorticity in the wake. The three-dimensional structure of dislocations is driven by the fact that over several shedding cycles, the out-of-phase von Kármán vortices must reconnect with the upstream and downstream shed vortex cycles from neighbouring cell(s) of different $\widehat {\textit {St}}_v$ . For the current forcing methodology, the spanwise constancy of $\widehat {\textit {St}}_v$ without low-frequency beating suggests a different wake structure from these studies. The work of Kim & Choi (Reference Kim and Choi2005) suggests that the reason for the inconsistency in the number of shedding frequencies among different studies may be due to the spanwise wavelength of the flow perturbation. They identified $\lambda _z=5d$ as a critical point below which $\widehat {\textit {St}}_v$ was constant along the span and above which the wake started to exhibit two shedding frequencies. The forcing wavelength of $2.4d$ adopted in the present study falls into the low-wavelength regime, so a constant $\widehat {\textit {St}}_v$ is consistent with most other studies in this regime. Naghib-Lahouti et al. (Reference Naghib-Lahouti, Hangan and Lavoie2015) is a notable outlier reporting different $\widehat {\textit {St}}_v$ for $\lambda _z=2.4d$ ; however, their $\widehat {\textit {St}}_v$ values were measured indirectly from the convective velocity and spatial separation of the von Kármán vortices, which may have introduced errors.

Measurements of the wake in the streamwise $\{x,y\}$ planes at $Re_d=2500$ are used to directly examine the separated shear layers and the vortex street. A representative instantaneous velocity/vorticity field without forcing is shown in figure 15(a). It exhibits the characteristic interaction of the separated shear layers and the resulting vortex street. The vortex formation length is approximately $1.5d$ . Hence, the von Kármán vortex in the phase shown at $x \approx 1d$ is still growing from the lower shear layer and the vortex that originates from the upper shear layer at $x \approx 2d$ has been pinched off by the fluid drawn from the lower part of the base region across the wake centreline.

The periodic component associated with vortex shedding is isolated by conditionally averaging the velocity field measurements with respect to the shedding phase. For the streamwise planes, the reference for the shedding cycle is obtained using a method based on proper orthogonal decomposition (POD) as detailed by van Oudheusden et al. (Reference van Oudheusden, Scarano, van Hinsberg and Watt2005). The snapshots were sorted into 18 phase bins, translating to roughly 85 snapshots per bin. Figure 15(b) plots the velocity/vorticity field for the unforced case averaged at a constant phase that roughly corresponds to that shown in figure 15(a). The basic symmetry of vortex shedding is evident with two rows of spanwise vortices that are $180^\circ$ out of phase between the $y\gt 0$ and $y\lt 0$ sides. Because vortex shedding is the dominant feature of the wake in the absence of forcing, the instantaneous and phase-averaged fields appear very similar.

Figure 15.

Vorticity field with velocity vectors in the streamwise plane without forcing at $Re_d=2500$ . (a) Sample instantaneous field. (b) Phase-averaged field with respect to the vortex shedding.

The results for $R=3.1$ at $Re_d=2500$ are examined in the streamwise plane because at this forcing amplitude, the jet vortical structures penetrate approximately up the edges of the boundary layers. This provides strong perturbations to the wake without causing the wake to significantly widen. Phase-averaged velocity/vorticity fields for this case in spanwise-separated planes at $z=\{0.08d, \: 0.24d, \: 0.38d, \: 0.6d \: \mathrm {and} \: 0.82d\}$ are shown in figure 16. Phase-averaging reveals vortex shedding in all the planes with the same basic symmetry as the unforced case. The shedding phases that are displayed are selected to be consistent with the phase shown in figure 15, approximately corresponding to when the negative vortex from the upper side has been fully shed from the body. This phase of the vortex shedding also aligns with that shown in figure 11(b) when the induced streamwise vortices in the separated shear layer reach maximum strength in the lower shear layer at $x=1d$ . It is emphasised, though, that this does not imply actual synchronisation between the measurements in the different planes. The vortices on either side of $y=0$ have approximately the same strength; however, the overall coherence and strength of the vortex street are reduced, and its three-dimensional nature is apparent from the significantly different distributions of the vorticity across the measurement planes.

Figure 16.

Phase-averaged vorticity fields with velocity vectors in different streamwise planes and forcing for $R=3.1$ at $Re_d=2500$ : (a) $z=0.08d$ , (b) $z=0.24d$ , (c) $z=0.38d$ , (d) $z=0.6d$ and (e) $z=0.82d$ .

In order to quantify the changes in the vortex shedding along the span, the downstream evolution of the circulation contained in the von Kármán vortices, $\varGamma _v$ , is computed. The vortices are identified here using the $Q$ -criterion (Hunt, Wray & Moin Reference Hunt, Wray and Moin1988), where $Q$ is the second invariant of the velocity-gradient tensor, $\nabla U$ , for incompressible flows. Mathematically, $Q=-(\partial u_i/\partial x_\kappa ) (\partial u_\kappa /\partial x_i)$ , which for 2-C planar PIV reduces to

(3.4) \begin{equation} Q = -\left ( \frac {\partial u}{\partial x} \right )^2 - 2 \left ( \frac {\partial u}{\partial y} \right ) \left ( \frac {\partial v}{\partial x} \right ) - \left ( \frac {\partial v}{\partial y} \right )^2, \end{equation}

if $w$ and the gradients in $z$ are neglected. According to the $Q$ -criterion, vortices exist for $Q \gt Q_{thresh}$ , where $Q_{thresh}$ is theoretically 0. A physical interpretation of this criterion is that vortices are regions where swirling motions make a larger contribution to the vorticity than irrotational shearing motions; however, it does not guarantee that every vortical structure is identified or that every contour of $Q \gt 0$ actually corresponds to a vortex. The circulation can be calculated by integrating the spanwise vorticity inside the vortical regions identified for each phase-averaged bin as

(3.5) \begin{equation} \frac {\varGamma _v}{u_\infty d} = \iint _{S} \left (\frac {\widetilde {\omega }_z d}{u_\infty } \right )\frac {1}{d^2} \mathrm {d}A , \end{equation}

and the centroids of the vortices are calculated as

(3.6) \begin{equation} x_v = \iint _{S} x \left (\frac {\widetilde {\omega }_z d}{u_\infty } \right ) \frac {1}{d^2} \left ( {\frac {\varGamma _v}{u_\infty d}} \right )^{-1} \mathrm {d}A , \end{equation}

(3.7) \begin{equation} y_v = \iint _{S} y \left (\frac {\widetilde {\omega }_z d}{u_\infty } \right ) \frac {1}{d^2} \left ( {\frac {\varGamma _v}{u_\infty d}} \right )^{-1} \mathrm {d}A . \end{equation}

Figure 17.

Mean downstream evolution of (a) the circulation contained in the von Kármán vortices and (b) their path in the different streamwise planes that originate from the lower shear layer of the body at $Re_d=2500$ without and with forcing at $R=3.1$ .

In practice, contours at $Q = 0$ are overly sensitive to measurement noise near the edges of the vortices, and so a slightly positive threshold is preferred to clearly define the vortex boundaries. Thresholds ranging from $0$ to $0.5$ were tested for $Q d^2/u_\infty ^2$ , and $0.01$ was selected for the current datasets. For relatively more positive $Q d^2/u_\infty ^2$ thresholds, the computed circulation values are generally lower because smaller vortical structures are identified. A complication with any $Q \geqslant 0$ threshold is that some regions of generally positive $Q$ may have a few points where $Q \lt 0$ . On occasion, this can result in separate $Q$ regions in close proximity to each other which are clearly part of the same overall vortex. In order to include the entirety of the vortex, the vorticity in the separate regions are added together when any part of their boundaries is within $0.18d \approx 0.05\overline {\lambda }_x$ of each other.

The streamwise evolution of the circulation and the paths of the von Kármán vortices are plotted in figure 17 for the unforced and $R=3.1$ cases at $Re=2500$ . In the absence of forcing, the circulation contained in a vortex just after it is shed is $\varGamma _v/(u_\infty d) = 1.5$ . The strength of the vortices reaches their maximum at this point and steadily decays downstream. With forcing at $R=3.1$ , the circulation of the von Kármán vortices across the span is reduced compared with the unforced case. In the region just downstream of vortex formation ( $1.5d \lt x \lt 2d$ ), the circulation of the von Kármán vortices is lowest in the $z=0.6d$ plane. This is because a significant fraction of the vorticity entering the wake in the $z=0.6d$ plane is not immediately available to be entrained into the von Kármán vortices since it is contained in the outer shear layers. As can be appreciated from figure 16, by $x \approx 2.5d$ , the outer shear layers are entrained into the von Kármán vortices in the $z=0.6d$ plane. This also occurs to an extent in the $z=0.38d$ plane; however, in that case, the vortical regions identified by the $Q$ method encompass both the inner and outer shear layer regions, and so all of this vorticity is summed with the $Q$ methodology. The von Kármán vortices approach the same strength in all the measurement planes for $x \gt 2.5d$ , containing about 65 % of the circulation of the unforced vortices by $x=5d$ .

The circulation that forms the vortex street originates from boundary layer separation at the trailing edges of the BTE-profiled body. The rate at which circulation is introduced into the wake from the separated shear layers, $\mathrm {d}\varGamma _s/\mathrm {d}t$ , can be calculated by integrating from the inner edge, $b$ , to the outer edge, $s$ , of the shear layer:

(3.8) \begin{equation} \frac {\mathrm {d}\varGamma _s}{\mathrm {d}t} = \int _b^s \overline {u} \, \overline {\omega }_z \mathrm {d}y = \int _b^s \overline {u} \frac {\partial \overline {u}}{\partial y} \mathrm {d}y = \frac {1}{2} \left (\overline {u}_s^2-\overline {u}_b^2 \right ). \end{equation}

Equation (3.8) approximates $\partial \overline {v}/\partial x=0$ , which should be valid for boundary layer flows and around the separation points. Furthermore, it is convenient to assume $\overline {u}_b=0$ because the fluid near the trailing edges inside the recirculation region is essentially stationary. Hence, the rate of circulation entering the wake is effectively determined by the excess velocity at the outer edge of the shear layer, $\overline {u}_s$ . It is instructive to relate the circulation supply from the shear layers, $\varGamma _s$ , and the resulting average circulation of the shed vortices. From one side of the body, $\varGamma _s$ can be estimated from the $\overline {u}$ profile near the separation point/trailing edge assuming that the flow is steady. Following the methodology of Roshko (Reference Roshko1954), this involves integrating (3.8) over a shedding period as

(3.9) \begin{equation} \Gamma _s = \int _0^{1/f_v} \frac {\mathrm {d}\varGamma _s}{\mathrm {d}t} \mathrm {d}t = \frac {\overline {u}_s^2}{2f_v}. \end{equation}

In the unforced $Re_d=2500$ case, $\widehat {\textit {St}}_v=0.25$ and $\overline {u}_s = 1.125u_\infty$ at $x=0.2d$ (see figure 10), so the normalised circulation supply during a shedding cycle is $\varGamma _s/(u_\infty d) \approx 2.5$ . For the $\{Re_d=2500, \; R=3.1\}$ case, the separated shear layers asymptotically approach the free stream far enough away from the body, but also exhibit an overshoot (figure 10). In the region between the jets, $\overline {u}_s = 1.11u_\infty$ and for the profiles closer to the jet centreline, $\overline {u}_s = 1.1u_\infty$ . Hence, the nominal supply of circulation in the $\{Re_d=2500, \; R=3.1\}$ case is $\varGamma _s/(u_\infty d) \approx 2.45$ – about the same as without forcing.

A simple model that relates the circulation in the wake to the separated shear layers is $\varGamma _s = \varGamma _v + \varGamma _{\mathscr {R}}$ , where $\varGamma _{\mathscr {R}}$ represents the circulation that is cancelled within the recirculation region. Partial cancellation of the circulation supply from a separated shear layer occurs due to its mixing with opposite-sign vorticity from the shear layer on the other side of the body. This is the main way that vorticity is usually lost in the near wake. For the unforced $Re_d=2500$ case, $\varGamma _s/(u_\infty d) = 2.5$ and $\varGamma _v/(u_\infty d) = 1.5$ . This implies that in the absence of forcing $\varGamma _{\mathscr {R}}/(u_\infty d) \approx 1$ , i.e. about $60\,\%$ of the circulation survives the vortex formation process and $40\,\%$ is cancelled in the recirculation region. This ratio is similar to that reported in other studies of bluff bodies, such as Cantwell & Coles (Reference Cantwell and Coles1983). For the $R=3.1$ forcing case, $\varGamma _v/(u_\infty d)$ appears to settle at approximately $(0.3{-}0.4)u_\infty d$ lower than the unforced case in the measured planes. Because the distance that the vortices are shed from the body is similar in the unforced and $R=3.1$ cases at $Re_d=2500$ , it is reasonable to assume that a similar amount of vorticity is cancelled in the formation region as in the unforced case, i.e. $\varGamma _{\mathscr {R}}/(u_\infty d) \approx 1$ . Hence, this implies that about $(0.3$ – $0.4)u_\infty d$ of the total circulation is missing, which is suspected to correspond to the partial diversion of the spanwise von Kármán vortices into streamwise-oriented loops.

An indication of the three-dimensional structure of the vortex street can be obtained from the path of the spanwise vortices in the different planes as plotted in figure 17(b). Without forcing at $Re_d=2500$ , the average path of the vortex street is roughly a straight line at about $0.15d$ away from the wake centreline. In all of the measured planes of the $\{Re_d=2500, \; R=3.1\}$ case, the von Kármán vortices are initially shed farther away from the wake centreline and take much different paths downstream. In the $z \leqslant 0.24d$ planes, the centroids of the vortices start at $y \approx 0.4$ and gradually advance towards the wake centreline, such that by $x=5d$ , they approach $y=0.25d$ . However, in the $z \geqslant 0.6d$ planes, the centroids of the vortices move in the opposite direction, away from the wake centreline, to just below $|y|=0.4d$ at the edge of the measurement window. Therefore, despite the similar strength of the von Kármán vortices across the span for $x \geqslant 2.5d$ , the cores of the shed vortices exhibit wavy undulations along the spanwise direction in an average sense in response to the imposed forcing disturbances.

Figure 18.

Mean streamwise vorticity field with ( $v,w$ ) vectors in the cross-plane at $x=4d$ with forcing for (a) $\{Re_d=2500, \; R=2.7\}$ , (b) $\{Re_d=2500, \; R=3.1\}$ , (c) $\{Re_d=2500, \; R=3.9\}$ , (d) $\{Re_d=5000, \; R=1.9\}$ and (e) $\{Re_d=5000, \; R=2.5\}$ .

3.3. Wake three-dimensional structure and the forcing amplitude

The three-dimensional structure of the forced wake can be directly examined using the cross-sectional view from the cross-plane at $x=4d$ . The spatial distribution of the mean streamwise vorticity and the velocity with forcing at $\{Re_d=2500; \; R=2.7, \; 3.1 \; \mathrm {and} \; 3.9\}$ and $\{Re_d=5000; \; R=1.9 \; \mathrm {and} \; 2.5\}$ is shown in figure 18. The general symmetry is the same for all these cases, with a pattern of alternating positive and negative streamwise vorticity along the span that is symmetric about the $y=0$ and $z=0$ planes, forming four vortex concentrations per forcing wavelength. The $-\omega _x$ vortices are in the $\{+z,+y\}$ and $\{-z,-y\}$ quadrants with respect to the jet slot centred at $z=0$ , and vice versa for the other quadrants. This matches the sense of rotation of the vortices at $x=1d$ in the $|y| \lt 0.45$ region (see figure 8), indicating that the organisation enforced on the vortex street during its initial development is maintained downstream. There is no signature of the edge vortices emitted by the actuators at $x=4d$ , presumably because those vorticity concentrations are entrained into the overall vortex street or dissipated. Notably, the strength of the streamwise vortices at $x=4d$ declines from $R=3.1$ to $R=3.9$ at $Re_d=2500$ and from $R=1.9$ to $R=2.5$ at $Re_d=5000$ . This aligns with the change in the jet–boundary layer interaction at these $Re_d$ previously observed at $x=1d$ in figure 8. Furthermore, the vortices in the $\{Re_d=2500, \; R=2.7\}$ case are relatively closer in strength to those in the $\{Re_d=5000, \; R=1.9\}$ case than $\{Re_d=5000, \; R=2.5\}$ . This is another indication that $R$ is not ideal for comparing across different $Re_d$ with the present methodology.

The streamwise vortices at $x=4d$ are reflective of the spanwise phase variations in the vortex shedding. To examine the connection of these vortices to the vortex shedding, the measurements are phase-averaged with respect to the shedding cycle. The streamwise plane measurements indicate that for forcing amplitudes where the jets do not penetrate beyond the boundary layer, $x=4d$ is far enough downstream that the vortex street has established a periodic structure, whereas for the higher-forcing-amplitude cases, the structure may not yet be fully established. The cross-plane measurements are phase-averaged directly with respect to the velocity near the outer edge of the shed vortices close to the free stream at $\{x,y\}=\{4d,1.25d\}$ using the methodology described in Appendix A. Each case was subdivided into 16 phase bins, resulting in approximately 65 snapshots per bin.

Phase-averaging shows that the counter-rotating streamwise vortex pairs alternate between both sides of the wake over a shedding cycle at $x=4d$ . Figure 19 captures the phase when the streamwise vorticity is at its greatest in the $y \lt 0$ half for $R=2.7$ at $Re_d=2500$ . This case is emphasised in this section because it most clearly demonstrates the organisation of the induced three-dimensional wake structure, but a similar organisation is evident in the $\{Re_d=2500, \; R=3.1\}$ case and it also appears in the $Re_d=5000$ cases. In spite of the jets being active from both sides of the body throughout the entirety of the shedding cycle, almost no streamwise vorticity is evident in the $y\gt 0$ half at this phase. Half a shedding period later, the situation flips and streamwise vorticity is maximised in the $y\gt 0$ half of the wake, except that the streamwise vortices have opposite sense of rotation. The fact that the streamwise vorticity coherently varies over the shedding cycle indicates that symmetric perturbations to the wake (about $y=0$ ) still result in an anti-symmetric wake structure due to presence of vortex shedding. It is consistent with the streamwise vortices at $x=4d$ being loops that originate from the von Kármán vortices. The phases where the streamwise vorticity is maximised are expected to occur when the vortex legs are most normal to the cross-plane, and should also correspond to when the $\tilde {\omega }_y$ components of the loops are at their weakest. The comparable magnitude of $\tilde {\omega }_x$ in the cross-plane to $\tilde {\omega }_z$ of the von Kármán vortices in the streamwise plane underscores that these loops are a deformation of the von Kármán vortices. The spanwise separation between the positive and negative streamwise vortex pairs is approximately $1.2d$ . Therefore, the spanwise wavelength of the vortex loops matches the $\lambda _z = 2.4d$ of the actuator spacing.

Figure 19.

Phase-averaged streamwise vorticity field with ( $v,w$ ) vectors at a phase where it is maximised in the $y \lt 0$ half of the cross-plane at $x=4d$ with forcing at an amplitude of $R=2.7$ at $Re_d=2500$ .

The vortical symmetry with the present distributed forcing methodology is the same across all the forcing amplitudes and $Re_d$ investigated. It is similar to that reported by Park et al. (Reference Park, Lee, Jeon, Hahn, Kim, Kim, Choi and Choi2006) for distributed forcing with tabs, and is consistent with the mean vorticity fields shown in the study of Zhang & Lee (Reference Zhang and Lee2005), for example. Although Zhang & Lee (Reference Zhang and Lee2005) did not directly investigate the shedding phase-dependent variations, it is reasonable to infer vortex looping based on their presented results. The streamwise vorticity fields resulting from the present forcing methodology are also superficially similar to the fields in the study of Bhattacharya & Gregory (Reference Bhattacharya and Gregory2018). However, they reported that the streamwise vorticity distribution was independent of the shedding phase and did not alternate across the $y=0$ plane. This suggests that the streamwise-induced vortices in the work of Bhattacharya & Gregory (Reference Bhattacharya and Gregory2018) were not loops of the von Kármán vortices. Additionally, Bhattacharya & Gregory (Reference Bhattacharya and Gregory2018) found that the streamwise vortices could change their symmetry depending on the forcing amplitude and spanwise wavelength. This highlights that different three-dimensional forcing schemes can also be beneficial for the drag reduction of two-dimensional bluff bodies. However, it is clear that the imposition of streamwise vorticity in the wake through spanwise-induced variations of the streamwise velocity is a common feature of different three-dimensional control strategies and is closely connected to the underlying mechanism. In discussing spanwise wavy wakes from a linear stability perspective, Hwang, Kim & Choi (Reference Hwang, Kim and Choi2013) found that von Kármán vortex shedding was stabilised due to sinusoidal base flow perturbations along the span. They argued that the stabilisation mechanism in their study proceeded in two steps. First, their imposed spanwise variation in the streamwise velocity field tilted the von Kármán vortices into the streamwise direction, resulting in the generation of streamwise vorticity. Next, the streamwise vortex loops interacted with the spanwise von Kármán vortices due to the spanwise shear in the base flow to attenuate the vortex shedding. This is essentially in line with the present experimental observations even though the perturbation to the flow is not purely sinusoidal along the span with the synthetic jet array and some streamwise vorticity is directly introduced into the wake by the actuators.

The phase-averaged structure of the vortex street in unforced and forced conditions can be visualised in three dimensions by stacking the two-dimensional phase-averaged stereo cross-planes. The phase dimension is converted to the $x$ spatial dimension using the $\lambda _x \approx 3.5d$ of the vortex street that is measured in the streamwise planes. This provides a three-dimensional approximation of the wake structure. Isosurfaces of the phase-averaged vorticity for the unforced and $R=2.7$ cases at $Re_d=2500$ are given in figure 20. The vorticity is coloured by the vector components at a constant level of $\omega _i d/u_\infty = 0.3$ . The spanwise components of the von Kármán vortices are coloured red and blue for positive and negative concentrations, respectively, which show the characteristic anti-symmetric structure of vortex shedding. The $|\omega _x|$ component is green and $|\omega _y|$ is purple. In the unforced case, the $|\omega _x|$ and $|\omega _y|$ surfaces do not form coherent structures in a phase-averaged sense. These components are connected to construct vortex loops in the $R=2.7$ case that repeat along the span with a wavelength matching the spacing of the actuators. The loops exist in the braid region of the vortex street where the strain rate of the flow is high, outside of the cores of the von Kármán vortices. For convenience, the loops may be referred to as streamwise vortices, but as they wrap around a von Kármán vortex, the loops reorient to be principally in the $y$ direction.

Figure 20.

Phase-averaged vorticity isosurfaces, coloured by vector component at a constant level of $\omega _i d/u_\infty =0.3$ for $Re_d=2500$ cases: unforced and (b) $R = 2.7$ . Red and blue surfaces correspond to positive and negative $\omega _z$ , respectively: Green is $\omega _x$ and purple is $\omega _y$ , which are joined together to form unified surfaces.

Figure 21.

Vortical structure of the wake with distributed forcing. Streamwise vortex loops are pulled out of the spanwise-oriented von Kármán vortices. This forms a line of vorticity along the streamwise direction that switches direction once per shedding wavelength.

An illustration of the vortical structure is given in figure 21. The principal direction of the von Kármán vortices is spanwise in the illustration, but loops are pulled out of each von Kármán vortex in the upstream direction symmetrically about the jet centreline planes ( $z=0$ ) and wrap around the neighbouring upstream von Kármán vortex shed from the opposite side of the body. The sense of rotation of the streamwise vortices is determined by that of the von Kármán vortices. The vortex legs seen in the $y \gt 0$ half of the cross-plane originate from the $-\omega _z$ von Kármán vortices, and vice versa for the vortex legs in the $y \lt 0$ half. A vortex leg that is pulled out of a von Kármán vortex in the upstream direction at a given $z$ location will have the opposite sense of rotation to the streamwise vortex leg that is pulled in the exact same direction from a von Kármán vortex shed from the opposite side of the body because the spanwise sense of rotation of the von Kármán vortices switches. Over a shedding period, the sense of rotation of a streamwise vortex at a given $z$ location therefore switches sign once. Equivalently, it forms a line of streamwise vorticity along the $x$ direction that periodically alternates between positive and negative once during each shedding wavelength. Hence, the vortex loops link together von Kármán vortices that rotate in opposite directions, forming a closed structure that repeats every half shedding wavelength.

The type of symmetry in the forced wake is also exhibited in some other natural vortical flows. For example, it is reminiscent of the three-dimensional structure predicted by Ashurst & Meiburg (Reference Ashurst and Meiburg1988) in a temporally evolving plane shear layer subject to a Kelvin–Helmholtz instability. This pattern was later simulated and experimentally produced by Lasheras & Choi (Reference Lasheras and Choi1988), Meiburg & Lasheras (Reference Meiburg and Lasheras1988) and Lasheras & Meiburg (Reference Lasheras and Meiburg1990) for shear layers, as well as the wake of a flat plate at $Re_d \approx 100$ . Lasheras & Choi (Reference Lasheras and Choi1988) excited a shear layer with small sinusoidal spanwise perturbations and observed the development of three-dimensionality in the resulting flow. They found that the Kelvin–Helmholtz instability developed first, producing spanwise vortices. Farther downstream, the perturbed vorticity on the braids of the spanwise vortices was stretched in the streamwise direction to form vortex loops. These streamwise vortices then induced waviness along the cores of the spanwise vortices through nonlinear interactions. Meiburg & Lasheras (Reference Meiburg and Lasheras1988) observed the tilting of spanwise von Kármán vortices in the wake of a flat plate at $Re_d \approx 100$ with spanwise perturbations. The vortical symmetry observed in these studies as well as the present work is also essentially the same as the self-sustaining mode A secondary instability in wakes described by, for example, Williamson (Reference Williamson1996).

Poncet et al. (Reference Poncet, Hildebrand, Cott and Koumoutsakos2008) argued for a connection between optimal drag reduction and the natural secondary instabilities of modes A and B because their numerical work identified that drag minimums occurred for spanwise perturbations near the wavelengths of these instabilities. Similar arguments have been put forth by Dobre et al. (Reference Dobre, Hangan and Vickery2006), Bhattacharya & Gregory (Reference Bhattacharya and Gregory2015) and Naghib-Lahouti et al. (Reference Naghib-Lahouti, Hangan and Lavoie2015). In the present study, the arrangement of the vortex loops in the forced wake arises because the actuators are located symmetrically about both sides of the BTE body. This keeps the induced loops from the upper and lower sides of the body approximately in line downstream of the jets. The fact that this arrangement aligns with the mode A vortices could be construed as supporting the hypothesis of Poncet et al. (Reference Poncet, Hildebrand, Cott and Koumoutsakos2008). However, the mode A instability is typically associated with the early stages of the wake transition regime ( $Re_d \lt 1000$ ) and not in the turbulent wake regime of the present work. Moreover, the spanwise wavelength of the mode A vortices is about $4d$ (Ryan, Thompson & Hourigan Reference Ryan, Thompson and Hourigan2005). The fact that this is much greater than the wavelength of the forcing indicates that, at the very least, matching the mode A instability is not required for distributed forcing.

The other secondary wake instability for this BTE body geometry, mode B, is not a reorientation of the von Kármán vortices at all. The mode B instability is speculated to develop in the braid region of the von Kármán vortices (Williamson Reference Williamson1996) or to be a three-dimensional instability of the separated shear layers (Brede, Eckelmann & Rockwell Reference Brede, Eckelmann and Rockwell1996). This mode persists throughout the turbulent wake regime, but its characteristic spanwise wavelength is approximately $1d$ and its vortical symmetry is different from that induced in the current investigation (Gibeau et al. Reference Gibeau, Koch and Ghaemi2018). For the mode A, the vortices form lines of the same-sign streamwise vorticity along $x$ instead of alternating every half-shedding cycle. It may be possible to induce this type of symmetry in the streamwise vorticity field by staggering the actuators along the span by $\lambda _z/2$ between the upper and lower sides of the BTE body. However, the vertical symmetry of the wake would likely be lost with a staggered arrangement of the actuators, i.e. the strength of the von Kármán vortices would be different from both sides of the body at a given $z$ location, and this is unlike the mode B instability. Therefore, it would be challenging to produce the overall symmetry of the mode B even with a staggered arrangement of actuators at a spanwise wavelength of $\approx 1d$ . The spanwise wavelength of the present forcing configuration matches that of Naghib-Lahouti et al. (Reference Naghib-Lahouti, Hangan and Lavoie2015), who perturbed the wake at $\lambda _z=2.4d$ in order to match the wavelength of a mode B $^\prime$ instability. However, recent work by Gibeau et al. (Reference Gibeau, Koch and Ghaemi2018) has cast significant doubt on whether a mode B $^\prime$ exists at all. Moreover, the overall topology of the mode B $^\prime$ would match the mode B if it does exist, with the streamwise vortices maintaining the same sense of rotation along $x$ at a given $z$ location (Naghib-Lahouti et al. Reference Naghib-Lahouti, Lavoie and Hangan2014). Hence, the vortical structures introduced by the present forcing arrangement would not be expected to align with the mode B $^\prime$ either. It is therefore improbable that distributed forcing depends on interactions with secondary instabilities.

Figure 22 shows the distribution of $\overline {k}= 0.5 ( \overline {u^\prime u^\prime }+\overline {v^\prime v^\prime }+\overline {w^\prime w^\prime } )$ in the cross-plane without and with forcing at $\{Re_d=2500; \; R=0, \; 2.7, \; 3.1 \; \mathrm {and} \; 3.9\}$ . As previously indicated in figure 14, the magnitude of the velocity fluctuations in the near wake is largely reflective of the strength of the vortex shedding. The total magnitude of the fluctuations is generally the highest along the wake centreline ( $y=0$ ) because it experiences the full extent of the cross-stream velocity variations induced by the von Kármán vortices shed from both sides of the body, and the $v$ fluctuations are typically stronger than the $u$ fluctuations in this region. Forcing causes $\overline {k}$ to decline throughout the cross-plane, primarily because of the attenuation of the vortex shedding. As a result of the spanwise-periodic nature of this forcing strategy, $\overline {k}$ is noticeably lower closer to the jet centreline at $z=0$ compared with that between the jets.

Figure 22.

Distribution of $\overline {k}$ in the cross-plane at $x=4d$ when $Re_d=2500$ : (a) without forcing, (b) $R=2.7$ , (c) $R=3.1$ and (d) $R=3.9$ .

The streamwise vortices in the wake induced by the synthetic jet array resemble those measured for a wavy cylinder by Zhang & Lee (Reference Zhang and Lee2005). The sense of rotation of the vortices is such that the spanwise regions downstream of the jet centrelines are analogous to the cylinder saddles, and the regions between jet slots correspond to the nodes. In essence, the synthetic jet forcing creates similar spanwise velocity gradients in the separated shear layers to those that naturally occur for bodies with wavy geometries in order to achieve a similar effect on the vortex shedding. The situation is also analogous because in the low- $\lambda _z$ regime of a wavy cylinder, Zhang & Lee (Reference Zhang and Lee2005) found that the minimum $\overline {k}$ in the wake downstream of the vortex formation region occurs in the saddle plane, and with the synthetic jet array, $\overline {k}$ is minimised along the jet centreline plane.

The distribution of $\overline {\omega }_z$ and $\overline {k}$ with the synthetic jet forcing scheme is also similar to that measured by Bhattacharya & Gregory (Reference Bhattacharya and Gregory2018) in their high amplitude forcing and $\lambda _z \geqslant 4d$ regime for distributed plasma actuators on a cylinder; however, they argued that the streamwise vortices did not alternate with the vortex shedding cycle. Synthetic jets were used as the actuator for distributed forcing on a cylinder by Cui, Feng & Liu (Reference Cui, Feng and Liu2020), who reported introducing streamwise vorticity into the wake, but there was not a clear symmetry of the wake vortical structures downstream of the vortex formation length, perhaps because the vortex shedding was more significantly disorganised than in the present case. Therefore, the work of neither Bhattacharya & Gregory (Reference Bhattacharya and Gregory2018) nor Cui et al. (Reference Cui, Feng and Liu2020) is directly comparable with the present forcing methodology.

Figure 23.

Peak phase-averaged streamwise circulation of the von Kármán vortex loops during a vortex shedding cycle as a function of $R$ for $Re_d=2500$ and $5000$ .

The circulation of the streamwise vortices is computed to test whether it can account for the missing circulation in the spanwise von Kármán vortices from the streamwise-plane PIV measurements. The PIV cross-plane at $x=4d$ is divided into quadrants bounding $0 \leqslant |z| \leqslant 1.2d$ and $0 \leqslant |y| \leqslant 1.05d$ which encompass each leg of the streamwise vortices. The vortices in each quadrant are tracked in a phase-averaged manner to measure how the streamwise circulation changes over the vortex shedding cycle. The vorticity inside each of the legs is integrated to compute the circulation for each phase bin, $\phi _v$ , as

(3.10) \begin{equation} |\varGamma _x (\phi _v)| = \iint _{\mathcal {S}} | \tilde {\omega }_x (\phi _v) | \mathrm {d}A = \sum _i \sum _\kappa | \tilde {\omega }_x (\phi _v) | \Delta y \Delta x, \end{equation}

where $\mathcal {S}$ is the surface boundary, $i$ and $\kappa$ are indices and $\Delta x = \Delta y = 0.039d$ . The integration bounds are set by lines of constant vorticity at $|0.15u_\infty d|$ to reject noisy data near the vortex edges. The two vortices located in the $y\gt 0$ half, as well as the vortices in the $y\lt 0$ half, form pairs consistent with a vortex loop structure. The phase with maximum circulation is typically the same for the vortices in a pair, and variations in the circulation are approximately $180^\circ$ out of phase between the loops located in the $y\gt 0$ and $y\lt 0$ halves of the wake. The average of the peak circulations in all four quadrants is given by $\overline {|\widehat {\varGamma }_x|}$ , where

(3.11) \begin{equation} \overline {|\widehat {\varGamma }_x|} = \frac {1}{4} \left ( |\widehat {\varGamma }_x|_{\ Upper \, left} + |\widehat {\varGamma }_x|_{\ Upper \, right} + |\widehat {\varGamma }_x|_{\ Lower \, left} + |\widehat {\varGamma }_x|_{\ Lower \, right} \right ). \end{equation}

The variation of $\overline {|\widehat {\varGamma }_x|}$ with $R$ at $Re_d=2500$ and $5000$ is summarised in figure 23. It can be seen that $\overline {|\widehat {\varGamma }_x|}/(u_\infty d)$ increases up to about $0.45$ when $Re_d=2500$ and 0.4 when $Re_d=5000$ . Recalling the discussion in § 3.2, $\overline {|\widehat {\varGamma }_x|}/(u_\infty d)=0.45$ is in line with the amount of spanwise circulation hidden from observation in the streamwise planes. This supports the hypothesis that the total circulation entering the wake is similar with and without forcing, but that about one-third of this circulation is in the streamwise/cross-stream directions in the best-case forcing scenario.

Consistent with the observations from figure 18, figure 23 shows that $\overline {|\widehat {\varGamma }_x|}/(u_\infty d)$ is maximised at $R=3.1$ when $Re_d=2500$ and at $R=1.9$ when $Re_d=5000$ . An indication of how the scaling of the forcing amplitude should be modified to achieve better collapse with $Re_d$ is that the $\overline {|\widehat {\varGamma }_x|}$ peak of the $Re_d=2500$ case occurs at about the same forcing amplitude that a significant amount of positive vorticity starts to appear at the upstream side of the jet slot, which is associated with the jet structure changing to a type that is more penetrating (see figure 6). At both $Re_d=2500$ and $5000$ , this critical forcing amplitude aligns with when the jet vortices penetrate just up to the edge of the boundary layer, suggesting that the forcing effectiveness is more closely linked to the jet penetration. The wall-normal penetration rate of a jet in a cross-flow is related to the momentum flux introduced by the jet. Using an empirical power law based on $R^\gamma \mathcal {D}$ , the penetration distance of a continuous jet emitted from a circular orifice is conventionally described as

(3.12) \begin{equation} \frac {y_j}{R^\gamma \mathcal {D}} = A \left (\frac {x_j}{R^\gamma \mathcal {D}} \right )^n, \end{equation}

where $A$ , $n$ and $\gamma$ are constants, $y_j$ is the wall-normal jet penetration distance relative to the model surface and $x_j$ is the streamwise distance from the jet origin (Mahesh Reference Mahesh2013; Berk et al. Reference Berk, Hutchins, Marusic and Ganapathisubramani2018). The scaling of synthetic jet penetration is complicated by their unsteady nature. Assuming that the synthetic jet formation criterion is met and that the vortices generated by a synthetic jet are not re-ingested into their cavities, the circulation initially emitted into a flow comes from the developed vortex rings. Using a boundary layer approximation, the instantaneous rate of circulation generation, $\varGamma _j$ , during the formation of a vortex ring from a synthetic jet slot is

(3.13) \begin{equation} \frac {\mathrm {d} \varGamma _j (t)}{\mathrm {d}t} \approx \frac {\partial u_j(t)}{\partial x} u_j(t) \mathrm {d}x \approx \frac {u_j(t)^2}{2}, \end{equation}

where $u_j(t)$ is the instantaneous centreline jet velocity (Glezer Reference Glezer1988). The circulation emitted from a synthetic jet varies periodically over the expulsion cycle, and it can be shown in non-dimensional form by integrating (3.13) over the period of an expulsion cycle, $T_j/2$ , that $\varGamma _j/(\mathcal {D} u_\infty ) \propto R^2/\textit {St}_j$ , where $\textit{St}_j = f_j \mathcal {D}/u_\infty$ (Berk & Ganapathisubramani Reference Berk and Ganapathisubramani2019). Using as a physical basis that the wall-normal velocity of a jet is proportional to the emitted circulation in a vortex ring, Berk & Ganapathisubramani (Reference Berk and Ganapathisubramani2019) argued that synthetic jet penetration should be a function of $\varGamma _j/(\mathcal {D} u_\infty ) \propto R^2/\textit {St}_j$ . Hence, they proposed scaling the penetration distance as

(3.14) \begin{equation} \frac {y_j/d}{(R^2/\textit {St}_j)^{0.6} } = A \left (\frac {x_j/d}{(R^2/\textit {St}_j)^{0.6} } \right )^{0.26}. \end{equation}

It is instructive to examine whether the scaling of Berk & Ganapathisubramani (Reference Berk and Ganapathisubramani2019) can model the jet penetration for the various cases in the current work. The primary interest here is $y_j/d$ when it is close to the trailing edge of the BTE model. If the actual path that the jet takes is not included in the model, then the $x_j/d$ dependence can be grouped with the $A$ constant to form $A^\dagger$ . This results in a final simplification of

(3.15) \begin{equation} y_j/d = A^\dagger (R^2/\textit {St}_j)^{n^\dagger }, \end{equation}

by rearranging (3.14).

Shear layer velocity profiles were measured at a fixed streamwise location of $x=0.2d$ with HWA to determine $y_j/d$ for a range of forcing amplitudes that the jet did not penetrate outside the boundary layer at $Re_d = {2500, \; 3700 \; \mathrm {and} \; 5000}$ (see figure 10 as an example for the $Re_d=2500$ and $R=3.1$ case). For these forcing amplitudes, the induced edge vortices remain close to the jet slot edges, producing the highest peaks in $\overline {u^\prime u^\prime }$ in the profiles taken at $0.4d \leqslant z \leqslant 0.6d$ . Therefore, $y_j/d$ is most appropriately identified from the shear layer profiles in this range. The $y_j/d$ location of the jet is defined here as the local peak in $\overline {u^\prime u^\prime }$ , although alternative definitions such as the local minimum in the mean velocity or local maximum in the velocity gradient could have been used (Smith Reference Smith2002). The $\overline {u^\prime u^\prime }$ peak is used to define $y_j/d$ because it was more reliable than the other methods at discriminating between the peak associated with the jet and the peak from the wake shear layer of the BTE body itself. Figure 24 summarises the computed $y_j/d$ for the different forcing amplitudes measured at $Re_d=2500, \; 3700 \; \mathrm {and} \; 5000$ . The wall-normal jet penetration datapoints follow a trend line that suggests that the model of Berk & Ganapathisubramani (Reference Berk and Ganapathisubramani2019) is appropriate for the present forcing methodology. The curve fit is best described as $y_j/d = (0.1 \pm 0.01) (R^2/\textit {St}_j)^{0.49 \pm 0.09}$ , where notably this $n^\dagger$ is consistent with the value of $0.44$ obtained by Berk & Ganapathisubramani (Reference Berk and Ganapathisubramani2019).

Figure 24.

Mean penetration distance, $y_j/d$ , of the vortical structures generated by forcing, measured at $x=0.2d$ and $0.4d \leqslant z \leqslant 0.6d$ , over a range of $Re_d$ and forcing amplitudes where the vortical structures remain inside the boundary layer. The dashed line shows the best fit through the datapoints: $y_j/d = 0.1 (R^2/\textit {St}_j)^{0.49}$ .

The streamwise circulation of the von Kármán vortex loops is scaled based on the jet penetration height using $(R^2/\textit {St}_j)^{0.5}$ , rounding to $n^\dagger =0.5$ . Figure 25 shows that $\overline {|\widehat {\varGamma }_x|}$ exhibits good correspondence with $(R^2/\textit {St}_j)^{0.5}$ for both $Re_d=2500$ and $5000$ . This indicates that $\overline {|\widehat {\varGamma }_x|}$ is better expressed as a function of the jet penetration height rather than $R$ or $C_\mu$ for the present forcing methodology. Evidently, the effectiveness of the present forcing scheme is more related to how the jet penetrates inside the boundary layer rather than the strength of the jet itself. Because the boundary layer thickness, $\delta$ , for a given chord length is related to $Re_d^{-0.5}$ , a jet at the same $R$ penetrates to a higher $y_j/\delta$ when $Re_d$ increases. This helps to explain why the strength of the induced streamwise von Kármán vortex loops for the same $R$ increases with $Re_d$ . However, the corresponding decrease in the boundary layer thickness with $Re_d$ also restricts the maximum strength of the jet that can be attained without penetrating past the boundary layer edge. This limits the ability of the present forcing scheme to redirect the spanwise von Kármán vortices into streamwise vortex loops and points to a lower maximum effectiveness at higher $Re_d$ for a given BTE body geometry with the present forcing strategy.

Figure 25.

Peak phase-averaged streamwise circulation of the von Kármán vortex loops during a vortex shedding cycle as a function of $(R^2/St_j)^{0.5}$ for $Re_d=2500$ and $5000$ .

3.4. Drag

Attenuated vortex shedding is typically associated with reduced drag due to the close connection between the concentration and formation distance of the shed von Kármán vortices, and the resulting mean and fluctuating forces acting on a bluff body. A planar momentum balance is employed with the PIV data to estimate the drag changes as a result of forcing. The main assumptions are that the incoming flow is uniform and steady, the only external force that acts on the control volume is the drag in the $x$ direction, the flow enters and exits the control volume only along the $\{y,z\}$ plane boundaries and the pressure in the free stream at the outflow plane is the same as at the inflow plane. Based on these assumptions, the equation for the drag coefficient from the velocity profile at the exit plane, $S_{out}$ , is derived in Appendix B as

(3.16) \begin{equation} c_d = \frac {2}{u_\infty ^2} \int _{S,{out}} \left ( u_\infty (\overline {u}^2 + \overline {w}^2)^{1/2} - \overline {u}^2 - \overline {u^\prime u^\prime } + \overline {v^\prime v^\prime } \right ) \mathrm {d} \left (\frac {y}{d}\right ). \end{equation}

For the streamwise 2C-PIV planes, the contribution of the $w$ term is neglected since it is not measured. The value of $c_d$ in these planes is computed as a function of $x$ and plotted in figure 26 for $Re_d=2500$ . It can be seen that $c_d$ declines when moving downstream. The lack of convergence in $c_d$ is most likely related to the assumption that the pressure at the inflow is the same as the free-stream pressure in the wake when, in actuality, the free-stream pressure is still recovering in the near wake. Hence, the computed $c_d$ values are inflated. The true $c_d$ for the present BTE-profiled body is unknown but $c_d=0.57$ was measured by Li, Bai & Gao (Reference Li, Bai and Gao2015) using force transducers for a BTE-profiled body where $r_{LE}=d$ , so the true $c_d$ for the present body is likely similar. The best estimate of $c_d$ that can be achieved with a momentum balance from the present data is from the downstream edge of the measurement domain at $x = 5.5d$ . Without measuring further downstream to improve the reliability of the computations, the $c_d$ values at $x=5.5d$ should provide enough information for comparison purposes. In the absence of forcing, $c_d=0.71$ at $x=5.5d$ . The local $c_d$ values in all the measured streamwise planes of the $\{Re_d=2500, \; R=3.1\}$ case are consistently lower than those of the baseline case, highlighting that this forcing condition reduces the drag. As suggested by the spanwise-variable nature of this forcing strategy, the local $c_d$ changes significantly along the span. The drag is lowest close to the jet centreline due to the greater degree of vortex street attenuation in this plane and gradually increases away from the jet centreline.

Figure 26.

Downstream evolution of $c_d$ in the unforced $Re_d=2500$ case and with forcing at an amplitude of $R=3.1$ across the different streamwise PIV planes.

The cross-plane data are used to determine the net $\langle c_d \rangle$ by spanwise averaging over a distance of one forcing wavelength for $-1.2d \lt z \lt 1.2d$ . The measurements in the cross-plane allow for a more complete analysis of $\langle c_d \rangle$ because of the much better spanwise resolution compared with the spanwise-separated streamwise planes. Additionally, the $\overline {w}$ term of $c_d$ in (3.16) can be included using the cross-plane data. For the forcing cases, the $\overline {w}$ term contributes to at most about $5\,\%$ of the local $c_d$ at the $z$ locations where the forcing induces the strongest $\overline {w}$ motions. There is a greater bias error in the drag computations made from the cross-plane than from the streamwise planes because the cross-plane is located even closer to the model than the $x=5.5d$ point that was available from the streamwise planes. The streamwise-plane data indicate that the $c_d$ values in the cross-plane at $x=4d$ are about $0.1$ – $0.15$ higher than they would be if they were measured at $x=5.5d$ . Nevertheless, the trends in how $\langle c_d \rangle$ varies between the forcing cases are not expected to change much if the cross-plane was located further downstream. Because the relative $\langle c_d \rangle$ trends are of more interest than the absolute values for this study, this inflation from the cross-plane is an acceptable compromise for the greatly improved spanwise resolution.

Considering the bias of the computed $c_d$ values, the results are summarised as ratios relative to the unforced case, $\langle c_{d,0} \rangle$ , in figure 27. It presents $\langle c_d \rangle /\langle c_{d,0} \rangle$ as a function of $(R^2/St_j)^{0.5}$ for the $Re_d=2500$ and $5000$ cases as the red dashed lines. The drag exhibits a minimum at both $Re_d$ that is about 25 % lower than that of the respective unforced cases. As understood by von Kármán (Reference von Kármán1912), the pressure drag of a bluff body is directly related to the strength of its vortex street. Physically, the greater the concentration of the shed vortices, the larger the kinetic energy of the wake. Because the energy to create the vortices comes from the work done by the relative movement of a body through a fluid, less work is done when the vortex street is weaker and hence the drag is lower. Therefore, the degree of drag reduction achieved with the current forcing methodology is in line with the significant attenuation of the vortex shedding strength.

Figure 27 plots $\langle c_d \rangle /\langle c_{d,0} \rangle$ alongside $\overline {|\widehat {\varGamma }_x|}$ of the von Kármán vortex loops as the blue solid lines. As in figure 25, plotting $\langle c_d \rangle /\langle c_{d,0} \rangle$ as a function of $(R^2/\textit {St}_j)^{0.5}$ provides a much better collapse between the two $Re_d$ cases than would be achieved with $R$ or $C_\mu$ . This provides further evidence that the effectiveness of the current forcing methodology is more closely related to the wall-normal penetration distance of the jet near the trailing edge rather than the emitted jet strength. The maximum net drag reduction for both $Re_d$ occurs at around $(R^2/\textit {St}_j)^{0.5} \approx 2.5{-}3$ , which also corresponds to when $\overline {|\widehat {\varGamma }_x|}$ is maximised. In agreement with the observations of Rodriguez (Reference Rodriguez1991), this correspondence supports the hypothesis that the drag reduction is proportional to the amount of spanwise vorticity realigned into streamwise vorticity. By extension, it also suggests that similar results can be achieved with this forcing scheme for other bluff-body geometries that feature vortex shedding, and can be used to provide guidance in future studies to design better actuators or forcing strategies.

Figure 27.

Spanwise-averaged drag ratio, $\langle c_d \rangle /\langle c_{d,0} \rangle$ (red), plotted alongside the streamwise circulation, $\overline {|\widehat {\varGamma }_x|}$ , of the von Kármán vortex loops (blue) as a function of forcing amplitude. The forcing amplitude is scaled based on the jet penetration distance, $(R^2/\textit {St}_j)^{0.5}$ .Here $Re_d=2500$ and $5000$ are represented by the circle and square datapoints, respectively.