4.1 Mean statistics

The mean streamwise velocity is shown in figure 3, which is represented in pseudo-colour with the mean streamlines superimposed on it. It is clear that the gap height has significant influence on the flow topology. In figure 3(a), the cylinder attaches to the wall ( $G/D=0$ ), the flow in this case resembles that behind a backward facing step. The flow past the upper side of the cylinder separates and forms a large separation bubble. The separation bubble reattaches to the wall at approximately $x/D=7$ . There is also a secondary bubble immediately behind the cylinder, which is only partly captured by the present measurement. As the cylinder leaves the wall, there is a gap flow under the cylinder (see figure 3 b, $G/D=0.25$ ), the gap flow acts like a jet coming out of a slot due to the flow acceleration in the contracting tunnel formed by the lower front surface of the cylinder and the wall. The gap flow soon deflects away from the wall and penetrates through the separation bubble, leading to the formation of two large separation bubbles, an upstream one behind the cylinder and a downstream one on the wall. The deflection of the gap flow is related to the pairing between the boundary layer on the wall and the lower shear layer from the cylinder, which will be shown in the following figure.

Figure 3.

Mean velocity in streamlines with the streamwise component in pseudo-colour. (a) $G/D=0$ , (b) $G/D=0.25$ , (c) $G/D=0.5$ , (d) $G/D=1.0$ , (e) $G/D=1.5$ , (f) $G/D=2.0$ , (g) $G/D=2.5$ , (h) $G/D=3.0$ .

In figure 3(c) at $G/D=0.5$ , the gap flow is much stronger, it deflects away from the wall and maintains considerable momentum when it makes contact with the upper shear layer of the cylinder, while for the case of $G/D=0$ .25 the gap flow is much weaker when it reaches the upper shear layer. The stronger gap flow brings more momentum into the recirculation region, consequently the separation bubble on the wall is reduced. It should be noted here that the contact between the gap flow and the upper shear layer implies a complex vortex interaction behaviour, which will be shown later. At $G/D=0.5$ , the separation bubble behind the cylinder contains two counter-rotating regions, indicating the probable onset of wake vortex shedding, although the bubble is asymmetric and tilts away from the wall. At $G/D=1.0$ (figure 3 d), the size of the separation bubble on the wall is significantly reduced and the asymmetry of the bubble behind the cylinder is still observable. When $G/D$ is increased to 1.5 (figure 3 e), the asymmetry of the separation bubble behind the cylinder is not obvious. Although no separation bubble is observed in the mean flow, by checking the instantaneous flow field (not shown here) it is found that a local separation occurs close to the wall in the form of a small patch of recirculation fluid. This local separation is periodical, which is associated with the formation of secondary vortices in the boundary layer. At $G/D=2.0$ , this local and periodical separation is still present very close to the wall in the instantaneous flow field. For $G/D=2.5$ and 3.0 (figure 3 g–h), no boundary layer separation can be observed in the mean flow or in the instantaneous flow field, and also secondary vortices are not observed. However, the growth of the boundary layer thickness downstream of the cylinder is obvious, indicating the transition still occurs under the wake perturbations.

Figure 4 shows the root mean square of the streamwise velocity fluctuation $u_{rms}$ (in pseudo-colour) and the mean spanwise vorticity $\unicode[STIX]{x1D714}_{z}^{\ast }$ (in line contours). At $G/D=0$ in figure 4(a), the statistics is similar to that of a backward facing step. The shear layer separation from the upper side of the cylinder can be observed. At the present Reynolds number the shear layer can roll up into vortices, but the vortices are irregular and erratic, indicating that there is no dominant frequency of the flow. $u_{rms}$ growth initiates from the shear layer and reaches a considerable level in the region where the vortex rolling up occurs. For $G/D=0.25$ in figure 4(b) the flow is quite different. Three shear layers can be found downstream of the cylinder. Aside from the upper shear layer from the cylinder, two more shear layers are also present due to the gap flow, one is the lower shear layer from the cylinder and the other the separated boundary layer. Instead of cancelling each other, these two shear layers deflect away from the wall together, which is the so-called shear layer ‘pairing’ or ‘coupling’ described in previous studies (Price et al. Reference Price, Sumner, Smith, Leong and Paidoussis2002; Sarkar & Sarkar Reference Sarkar and Sarkar2010). It should be kept in mind that the vorticity for upper shear layer and the separated boundary layer are negative (clockwise direction) while the vorticity for the lower shear layer is positive (counter-clockwise direction), so a single patch of vorticity after $x/D>4$ indicates that there may be some kind of merging process of the vorticity when these three regions of concentrated vorticity move close to each other. For $G/D=0.5$ in figure 4(c) the vorticity distribution is similar to that for $G/D=0.25$ , also similar is the active (high $u_{rms}$ value) wake region and the relatively calm near-wall region. However, the separated boundary layer seems to be much stronger than that for $G/D=0.25$ .

Figure 4.

Mean vorticity $\unicode[STIX]{x1D714}_{z}^{\ast }$ in line contours with root mean square of the streamwise fluctuation $u_{rms}$ in pseudo-colour. $\unicode[STIX]{x1D714}_{z}^{\ast }$ is the normalized mean spanwise vorticity $\unicode[STIX]{x1D714}_{z}^{\ast }=\unicode[STIX]{x1D714}_{z}D/U_{\infty }$ . $u_{rms}$ is normalized by free-stream velocity $U_{\infty }$ . (a) $G/D=0$ , (b) $G/D=0.25$ , (c) $G/D=0.5$ , (d) $G/D=1.0$ , (e) $G/D=1.5$ , (f) $G/D=2.0$ , (g) $G/D=2.5$ , (h) $G/D=3.0$ .

For $G/D=1.0$ in figure 4(d), the deflections of the lower shear layer and the separated boundary layer are significantly reduced. The distribution of mean vorticity from the two sides of the cylinder is close to symmetry. The distribution of $u_{rms}$ , however, is quite asymmetric that the high value region ( $u_{rms}/U_{\infty }>0.25$ ) for the lower branch extends to approximately $x/D=7$ while that for the upper branch only reaches $x/D=4$ . As will be shown later, the larger high $u_{rms}$ region is closely related to the interaction between the lower wake vortex and the secondary vortex in the boundary layer. Further increase of $G/D$ makes the wake and boundary layer separately identifiable, as can be seen from the high $u_{rms}$ region near the wall in figure 4(e–h). Although the cylinder is already out of the camera’s field of view for high gap ratios, it is not difficult to deduce that as the cylinder moves away from the wall, the cylinder wake flow would be more and more like that of a stand-alone cylinder.

Figure 5.

Streamwise velocity spectra for different gap ratios. The velocity traces were taken from the PIV data at a streamwise location of $x/D=3.0$ , and the wall-normal locations are $y/D=1.35$ , 1.0, 1.15, 1.18, 1.6, 2.08, 2.0 and 2.0 for $G/D=0$ , 0.25, 0.5, 1.0, 1.5, 2.0, 2.5 and 3.0, respectively. For $G/D\leqslant 2.0$ , the chosen wall-normal locations correspond to local maxima of $u_{rms}$ , while for $G/D=2.5$ and 3.0, a maximum of $u_{rms}$ in the wake region is not available and $y/D=2.0$ is used instead. The frequency is normalized by $f_{0}$ , where $f_{0}$ is the wake vortex shedding frequency at $G/D=3.0$ .

The streamwise velocity spectra for different gap ratios are shown in figure 5. The velocity traces are taken at the streamwise location of $x/D=3.0$ , and the wall-normal locations are chosen to be at the local maxima of $u_{rms}$ which vary with $G/D$ . For $G/D\geqslant 0.5$ there is a dominant peak on the spectrum, which indicates the frequency of wake vortex shedding from the cylinder. For a fixed $G/D$ , this characteristic frequency does not change when the probe is placed in a number of different locations in the flow, meaning that the characteristic frequency of the vortex dynamics remained unchanged. However, figure 5 shows that with the decrease of $G/D$ , the characteristic frequency increases. For $G/D=0$ and 0.25, where the lower shear layer from the cylinder is inhibited or completely suppressed, no distinguished peaks can be observed on the spectra.

The dominant frequencies determined in figure 5 is plotted against the gap ratio in figure 6. As shown in figure 6 the Strouhal number $St$ ( $St=fD/U_{\infty }$ ) increases with decreasing gap ratio. The variation of $St$ is fitted very well by an exponential curve: $St=0.16\exp (-1.34G/D)+0.21$ , where 0.21 is the $St$ for a stand-alone circular cylinder. Price et al. (Reference Price, Sumner, Smith, Leong and Paidoussis2002) has already reported the change of $St$ as the cylinder approaches the wall at low Reynolds number ( $Re_{D}<2600$ ) and that $St$ was insensitive to $G/D$ at higher Reynolds number. The present $Re_{D}$ is in their low Reynolds number range and the variation of $St$ is quite considerable, from 0.22 at $G/D=3.0$ to 0.29 at $G/D=0.5$ , with nearly a 32 % increase. The $St$ in the present study compares well with that of Price et al. (Reference Price, Sumner, Smith, Leong and Paidoussis2002) for $G/D\geqslant 1.0$ . At $G/D=0.5$ , however, the difference is significant. Moreover, below $G/D=0.25$ , no dominant frequency in the spectra can be found in present study, while Price et al. (Reference Price, Sumner, Smith, Leong and Paidoussis2002) claimed that there did not seem to be a minimum value of $G/D$ below which periodicities were not detected in the wake. The $St$ in the study of Price et al. (Reference Price, Sumner, Smith, Leong and Paidoussis2002) can be as high as approximately 0.4 at $G/D=0.125$ . Price et al. (Reference Price, Sumner, Smith, Leong and Paidoussis2002) explained that it was more properly associated with periodicities in the outer shear layer from the cylinder instead of classical vortex shedding. In the present study, however, such periodicities in the outer shear layer cannot be found. This difference for low gap ratios between the present study and that of Price et al. (Reference Price, Sumner, Smith, Leong and Paidoussis2002) may be attributed to the different boundary layer thicknesses at the location of the cylinder ( $\unicode[STIX]{x1D6FF}/D=0.53$ in the present study and $\unicode[STIX]{x1D6FF}/D=0.45$ in their study). Price et al. (Reference Price, Sumner, Smith, Leong and Paidoussis2002) did not give an explanation for the change of $St$ at low Reynolds number, which can be expected to be strongly related to the vortex dynamics and will be explored in the present study.

Figure 6.

Vortex shedding frequency at different gap ratios. $St=fD/U_{\infty }$ . The fitted line is expressed as $St=0.16\exp (-1.34G/D)+0.21$ and only the data in the present study are taken into account.

Figure 7 shows the disturbance growth inside the boundary layer for $G/D\geqslant 1.5$ . As shown in figure 4, for these gap ratios the boundary layer can be identified separately. For $G/D<1.5$ , due to the separation of the boundary layer from the wall, the amplification of the disturbance occurs in the separated shear layer, which extends into the wake and results in turbulence generation mainly in the wake region (see figure 4). When the cylinder is absent, the disturbance in the laminar boundary layer varies little and is at about the level of the background noise. With the presence of the cylinder, disturbance growth will initiate downstream of the cylinder due to the transition triggered by the wake perturbations. The closer the cylinder is to the wall, the faster the disturbance growth will be. For the cases of $G/D=1.5$ and 2.0, the transition is featured by the evolution of secondary vortices in the boundary layer, a transition route that has been investigated by previous researchers, such as Pan et al. (Reference Pan, Wang, Zhang and Feng2008), Mandal & Dey (Reference Mandal and Dey2011) and He et al. (Reference He, Wang and Pan2013, Reference He, Pan, Feng, Gao and Wang2016) (which has been described in the introduction). He et al. (Reference He, Wang and Pan2013) pointed out that the fast disturbance growth in the boundary layer is related to the three-dimensional destabilization of the initially nearly two-dimensional spanwise secondary vortices. The disturbance reaches a peak at around $x/D=7$ and $x/D=10$ for $G/D=1.5$ and $G/D=2.0$ , respectively. Afterwards the disturbance begins to decay when the flow becomes highly nonlinear. The lack of data points for $G/D=1.5$ downstream of $x/D=8.5$ is due to the merging of the boundary layer with the wake.

Figure 7.

Disturbance growth in the boundary layer; $u_{rms.max}$ is the maximum of $u_{rms}$ in the boundary layer.

For $G/D=2.5$ and 3.0, the disturbance growth is slower than the previous two cases, but still much faster than the case of smooth boundary layer without the presence of the cylinder. As we have mentioned earlier, there are no secondary vortices in the boundary layer and the transverse expansion of the turbulence in the wake has not reached the boundary layer yet in the measured field of view for these two cases, so the transition of the boundary layer may well be attributed to the influence of an adverse pressure gradient. In the experiment of Bearman & Zdravkovich (Reference Bearman and Zdravkovich1978), they found that the presence of the circular cylinder in proximity to the wall would introduce a minimum of the pressure coefficient on the wall downstream of the cylinder. This means an adverse pressure gradient (APG) further downstream, and the closer the cylinder is to the wall, the higher the APG will be. Therefore, in the present study, when the cylinder is close enough to the wall ( $G/D<1.5$ ), the large APG leads to a constant separation of the boundary layer. When the gap ratio is moderate ( $G/D=1.5$ and $G/D=2.0$ ), the moderate APG (which is fluctuating with the wake) can only periodically induce a local separation of the boundary layer, leading to the formation of secondary vortices which convect downstream close to the wall. When the gap ratio is increased to $G/D=2.5$ and 3.0, the APG may be too weak to induce any separation, however, the APG is known to make the boundary layer more unstable and facilitates transition, therefore boundary layer disturbance growth is observed downstream of the cylinder.

Figure 8.

Phase-averaged $\unicode[STIX]{x1D706}_{ci}$ field, $G/D=0.5$ . Positive (counter-clockwise rotating) vortices are in red colour and solid contour lines, while negative (clockwise rotating) vortices are in blue colour and dashed contour lines, the lowest contour level is $\unicode[STIX]{x1D706}_{ci}=1$ and increment between contour lines is 1. (a) $\unicode[STIX]{x1D6F7}=0^{\circ }$ , (b) $\unicode[STIX]{x1D6F7}=90^{\circ }$ , (c) $\unicode[STIX]{x1D6F7}=180^{\circ }$ , (d) $\unicode[STIX]{x1D6F7}=270^{\circ }$ . See supplementary movie 1 available online at https://doi.org/10.1017/jfm.2016.812.

4.2 Phase-averaged vortical structures

This section presents the spanwise vortical structures in phase-averaged flow field for three cases of $G/D=0.5$ , 1.0 and 2.0, which are representative of vortex dynamics in cylinder wake/boundary layer interactions. For $G/D>2.0$ , no vortical structures (boundary layer separation) is observed in the near-wall region downstream of the cylinder and the wake vortices of the cylinder are just like that of a stand-alone cylinder. For $G/D<0.5$ , no clear periodicity can be determined, therefore phase averaging is not possible. In performing the phase averaging, the frames at a certain phase angle are determined with a cross-correlation between all the velocity frames with the base frame and then selecting the ones corresponding to the correlation peaks. These selected frames are averaged to yield the flow field at that phase angle. This step is repeated until averaged flow fields are extracted at all the needed phase angles. The vortices are identified from the phase-averaged flow field by the $\unicode[STIX]{x1D706}_{ci}$ method proposed by Zhou et al. (Reference Zhou, Adrian, Balachandar and Kendall1999). $\unicode[STIX]{x1D706}_{ci}$ is the imaginary part of the complex eigenvalue of the velocity gradient tensor. In the present study, $\unicode[STIX]{x1D706}_{ci}$ is computed from the two-dimensional truncated velocity gradient tensor due to the limitation of two-dimensional PIV measurement. The sign of the local vorticity is assigned to the $\unicode[STIX]{x1D706}_{ci}$ to indicate the rotation sense.

Figure 9.

Phase-averaged $\unicode[STIX]{x1D706}_{ci}$ field, $G/D=1.0$ . (a) $\unicode[STIX]{x1D6F7}=0^{\circ }$ , (b) $\unicode[STIX]{x1D6F7}=90^{\circ }$ , (c) $\unicode[STIX]{x1D6F7}=180^{\circ }$ , (d) $\unicode[STIX]{x1D6F7}=270^{\circ }$ . See supplementary movie 2.

In figure 8 at $G/D=0.5$ , phase-averaged results show that the alternative wake vortex shedding has already been restored. However, the shedding is asymmetric about the centreline of the cylinder, the upper wake vortex moves nearly parallel with the wall while the lower wake vortex moves away from the wall and even surpasses the height of the upper one. This result is what Sarkar & Sarkar (Reference Sarkar and Sarkar2010) described as cross-over of the vortex trajectories and inversion of the vortex positions, with the wake vortex from the lower side of the cylinder being above the one from the upper side. During this cross-over, the lower wake vortex shrinks very fast and soon loses its trace. Aside from the two wake vortices, there is another vortex below the lower one. This vortex is formed by the rolling up of the separated boundary layer that pairs with the lower shear layer. It rotates in the opposite direction to the lower wake vortex and has the same shedding frequency as the latter. We denote it the secondary vortex as it can be regarded as being induced by the passing wake vortices, as a result of vortex-induced instability of wall-bounded shear layers (Sengupta, De & Sarkar Reference Sengupta, De and Sarkar2003; Dipankar & Sengupta Reference Dipankar and Sengupta2005). The separated boundary layer becomes turbulent as it gradually merges into the wake (see figure 4 c), and the flow finally reattaches onto the wall, forming a separation bubble (see figure 3 c). It should be emphasized that the secondary vortex is an instantaneous vortical structure while the separation bubble is based on the mean streamlines. The separation bubble, although having closed or spiral mean streamlines, does not represent the secondary vortex which is much smaller in size and convects with the flow. In figure 8 the secondary vortex moves away from the wall together with the lower wake vortex. When it moves close to the upper wake vortex it seems to merge with the latter and lose its identity very quickly. The above results suggest that at $G/D=0.5$ , these three vortices come very close to each other and have direct interactions among them. The upper wake vortex and the secondary vortex have the same rotation sense and seem to merge together while the lower wake vortex may be canceled because it has the opposite sign and is relatively weaker than the sum of the other two vortices. Consequently, only negative vortices (clockwise rotating) are observed in the downstream wake.

Figure 10.

Phase-averaged $\unicode[STIX]{x1D706}_{ci}$ field, $G/D=2.0$ . (a) $\unicode[STIX]{x1D6F7}=0^{\circ }$ , (b) $\unicode[STIX]{x1D6F7}=90^{\circ }$ , (c) $\unicode[STIX]{x1D6F7}=180^{\circ }$ , (d) $\unicode[STIX]{x1D6F7}=270^{\circ }$ . See supplementary movie 3.

In figure 9 at $G/D=1.0$ , the three vortices in the previous case are still present. The lower wake vortex still moves away from the wall, but it only gets elevated to the same height as the upper wake vortex. At this gap ratio, the lower wake vortex convects for a longer streamwise distance than that at $G/D=0.25$ , but still decays faster than the upper wake vortex. The lift-up of the secondary vortex is also reduced and it can no longer interact with the upper wake vortex. Alternatively, the interaction at this gap ratio is mainly between the secondary vortex and the lower wake vortex, which cannot be seen from this phase-averaged result. One previous study of He et al. (Reference He, Wang and Pan2013) at $G/D=1.0$ has revealed through visualization that after the secondary vortex is formed in the near-wall region and gets lifted up, it directly interacts with the lower wake vortex after $x/D=4$ . Because the secondary vortex has the opposite rotation sense to the lower wake vortex and is weaker than the latter, it ends up either being rolled up into the wake or being pushed down to the wall, depending on its relative position with the lower wake vortex. This interaction results in the fast breakdown of the secondary vortex which can be observed in the near-wall region of figure 9. The interaction also results in an increased turbulence level in the related region and causes the asymmetry of the turbulent statistics ( $u_{rms}$ in figure 4 d). Although the lower wake vortex is less affected by this interaction, the increased turbulence level facilitates its breakdown into small vortices and therefore it appears to decay faster than the upper wake vortex.

In figure 10 at $G/D=2.0$ , the secondary vortex and the lower wake vortex are shown, as well as part of the upper wake vortex. It can be seen that the lower wake vortex moves nearly parallel with the wall and so does the secondary vortex. In this case, the distance between the secondary vortex and the lower wake vortex prevents any direct interactions from happening. However, the wake provides perturbations that can promote the destabilization of the secondary vortices. The evolution of the secondary vortices has been investigated by recent studies of Pan et al. (Reference Pan, Wang, Zhang and Feng2008) and He et al. (Reference He, Wang and Pan2013, Reference He, Pan, Feng, Gao and Wang2016). It has been revealed that the spanwise secondary vortex develops three-dimensional instability under the perturbation of the wake and deforms into $\unicode[STIX]{x1D6EC}$ vortices, which further evolve into hairpin vortices. The hairpin vortices can generate new hairpin vortices in a hierarchy, leading to the formation of hairpin packets in the final turbulent boundary layer.

Figure 11.

Trajectories of vortices. (a) $G/D=0.5$ , (b) $G/D=1.0$ , (c) $G/D=1.5$ , (d) $G/D=2.0$ .

The trajectories of the large spanwise vortices in the phase-averaged flow field are shown in figure 11 for $G/D=0.5$ , 1.0, 1.5 and 2.0. The vortex centres are identified as the weighted centroid of the $\unicode[STIX]{x1D706}_{ci}$ concentration region with threshold of $\unicode[STIX]{x1D706}_{ci}^{\text{T}}=1.0$ . As discussed in the previous section, the gap ratio has a huge impact on the trajectories of lower wake vortex and the secondary vortex, while the trajectory of the upper wake vortex is little affected and nearly parallel to the wall for all gap ratios tested. At $G/D=0.5$ , the lift-up of the secondary vortex and the lower wake vortex is quite obvious and the trajectories of the upper and lower wake vortices clearly cross each other, resulting in the inversion of their positions, as mentioned above. The lower wake vortex and the secondary vortex lose their traces very fast. This is due to their interactions with the upper wake vortices which will be shown later in detail by the instantaneous vortical structures. At $G/D=1.0$ , the angles of the secondary vortex and the lower wake vortex convecting away from the wall are reduced. The lower wake vortex can reach the height of that of the upper wake vortex but there is no cross-over of the trajectories. The lower wake vortex in this case persists longer but still ‘diminishes’ faster than the upper wake vortex. With the increased gap ratio the deflection angles further decrease (see figure 11 c,d). By $G/D=2.0$ , trajectories of both the secondary vortex and the lower wake vortex are nearly parallel to the wall, the trajectory of the upper wake vortex cannot be given because it is only partly captured in the present field of view (see figure 10).

A direct comparison of secondary vortex trajectories at different gap ratios is presented in figure 12(a). The initial lift-up of the secondary vortex trajectories seems to be a linear increase and is fitted with straight lines to yield the deflection angles. The deflection angles for the cases of $G/D=0.5$ , 1.0, 1.5, 2.0 are $25^{\circ }$ , $11^{\circ }$ , $6^{\circ }$ and $1^{\circ }$ , respectively, showing the large variation in convection direction of the secondary vortex. Figure 12(b) gives the normalized circulation $\unicode[STIX]{x1D6E4}^{\ast }$ of the secondary vortices for the above four cases. Downstream of the cylinder, the secondary vortex forms in terms of accumulation of vorticity from the boundary layer, so a continual growth of the $\unicode[STIX]{x1D6E4}^{\ast }$ can be observed. The formation process can be deemed complete when $\unicode[STIX]{x1D6E4}^{\ast }$ reaches its peak. The decrease of $\unicode[STIX]{x1D6E4}^{\ast }$ afterwards is related to the further development of the secondary vortices, including three-dimensional deformation or even disintegration. It can be observed from figure 12(b) that as the cylinder approaches the wall ( $G/D=2.0$ , 1.5 and 1.0) but stays out of the boundary layer ( $\unicode[STIX]{x1D6FF}/D=0.53$ ), the growth of $\unicode[STIX]{x1D6E4}^{\ast }$ initiates earlier and the peak value is larger, which is the result of stronger interaction between the cylinder wake and the boundary layer. The maximum $\unicode[STIX]{x1D6E4}^{\ast }$ can be found for $G/D=1.0$ . At $G/D=0.5$ , however, the secondary vortex is weaker than the case of $G/D=1.0$ , this is because the cylinder is already positioned in the outer part of the boundary layer and the lower shear layer from the cylinder is suppressed, therefore the induced secondary vortex is weaker. It should be noted that at $G/D=0.5$ the secondary vortex loses its trace very early near $x/D=4$ without a continual decrease as in the higher gap ratio cases, which is due to its interaction with the upper wake vortex (shown in the following).

Figure 12.

(a) Trajectories and (b) circulation of secondary vortices. $\unicode[STIX]{x1D6E4}^{\ast }$ in (b) is the normalized circulation, $\unicode[STIX]{x1D6E4}^{\ast }=\unicode[STIX]{x1D6E4}/(\unicode[STIX]{x03C0}DU_{\infty })$ .

4.3 Instantaneous vortical structures

The formation and convection of vortices shown above are in phase-averaged flow, where the turbulent fluctuation and small-scale structures are smoothed out to allow attention to be paid to large periodical structures. The results also show indications of vortex interaction when they move close enough to each other for low gap ratios such as $G/D=0.5$ . The interaction often leads to the final disintegration or breakdown of vortices, which can only be shown in detail in the instantaneous flow field with a proper visualization method. At the Reynolds number in the present study, the wake is turbulent, therefore the vorticity plot of a vortex in the instantaneous flow is irregular in shape and may emerge in pieces before interaction (see Sarkar & Sarkar Reference Sarkar and Sarkar2010). Moreover, it is almost impossible to distinguish the vorticity of two same-signed vortices as soon as they make contact. Therefore, the vorticity plot is perhaps not an appropriate way to investigate vortex interactions in the present study and an alternative method of virtual dye visualization is applied instead.

In the present study, we use virtual dye visualization with a particular purpose: to reveal the complex vortex interactions. The virtual dye is released in the upper shear layer, lower shear layer and the boundary layer at the same time, the corresponding upper wake vortex, the lower wake vortex and the secondary vortex are then visualized, respectively.

Figure 13.

Instantaneous vortical structures, $G/D=0.5$ . (a) $t$ , (b) $t+\unicode[STIX]{x0394}t$ , (c) $t+2\unicode[STIX]{x0394}t$ , (d) $t+3\unicode[STIX]{x0394}t$ , $\unicode[STIX]{x0394}t/T_{0.5}=0.3$ , where $T_{0.5}$ is the vortex shedding period for $G/D=0.5$ . The upper wake vortex, the lower wake vortex and the secondary vortex are tagged in blue, cyan and red colours, respectively. See supplementary movie 4.

Figure 14.

Instantaneous vortical structures, $G/D=1.0$ . (a) $t$ , (b) $t+\unicode[STIX]{x0394}t$ , (c) $t+2\unicode[STIX]{x0394}t$ , (d) $t+3\unicode[STIX]{x0394}t$ , $\unicode[STIX]{x0394}t/T_{1.0}=0.26$ , where $T_{1.0}$ is the vortex shedding period for $G/D=1.0$ . The upper wake vortex, the lower wake vortex and the secondary vortex are tagged in blue, cyan and red colours, respectively. See supplementary movie 5.

At $G/D=0.5$ in figure 13, the formation of upper wake vortex, the lower wake vortex and the secondary vortex are clearly revealed. Our focus here is on the subsequent interaction among them. As mentioned earlier, vortex interaction occurs when the lower wake vortex and the secondary vortex move up towards the upper wake vortex (as shown by the circled area). One should remember that the upper wake vortex and the secondary vortex have the same sign and rotate in the clockwise direction while the lower wake vortex has the opposite sign. It can be observed in figure 13 that the three vortices are all stretched to some extent, but the lower wake vortex is stretched the most. Besides, the lower wake vortex and the secondary vortex are gradually encompassed by the upper wake vortex and its tail, which is usually a thin shear layer following a shed wake vortex. During the interaction process, the lower wake vortex is under severe stretching while the upper wake vortex and the secondary vortex can maintain their vortical shapes and rotate about each other, typically seen in the merging of two co-rotating vortices in a mixing layer which is called ‘vortex pairing’ (Winant & Browand Reference Winant and Browand1974). In figure 13(d) the lower wake vortex has been stretched almost into a thin layer and can be regarded as having already broken down, a process similar to the ‘vortex tearing’ in a mixing layer (Moore & Saifman Reference Moore and Saifman1975; Saffman & Baker Reference Saffman and Baker1979), where the vorticity of one vortex is entrained and absorbed into the other vortices. In figure 13(d) the upper wake vortex and the secondary vortex nearly switch their relative positions as compared with figure 13(a), they go through further deformation and seem to penetrate into each other, a stage which can be more clearly revealed by the late-stage vortex interaction from the previous shedding period in the downstream region of $x/D=7{-}8$ in figure 13(b,c). This indicates the merging of these two same-signed vortices. So the result of the vortex interaction in this case is the merging of upper wake vortex and the secondary vortex, and the breakdown of the lower wake vortex. This explains why only clockwise rotating vortices are detected downstream. The terms ‘vortex tearing’ and ‘vortex pairing’ are two mechanisms proposed to explain the disappearance or disintegration of vortices in mixing layers and obviously they both have particular implications in the present complex flow scenario.

Figure 15.

Instantaneous vortical structures, $G/D=2.0$ . (a) $t$ , (b) $t+\unicode[STIX]{x0394}t$ , (c) $t+2\unicode[STIX]{x0394}t$ , (d) $t+3\unicode[STIX]{x0394}t$ , $\unicode[STIX]{x0394}t/T_{2.0}=0.3$ , where $T_{2.0}$ is the vortex shedding period for $G/D=2.0$ . The lower wake vortex and the secondary vortex are tagged in blue and red colours, respectively. See supplementary movie 6.

At $G/D=1.0$ in figure 14, the lower wake vortex and the secondary vortex are not lifted up as high as that in $G/D=0.5$ , they keep a certain distance from the upper wake vortex. In this case the interaction is mainly between the opposite-signed lower wake vortex and the secondary vortex. As the lower wake vortex is stronger, a large portion of the secondary vortex is quickly stretched into a thin layer, and gets rolled up into the wake vortex as mentioned before. Dritschel (Reference Dritschel1995) theoretically investigated the two-dimensional vortex interactions and found the filamentation of vortices between both same-signed and opposite-signed vortices with equal vorticity but unequal sizes. The external flow perturbation to the vortices is assumed to be weak, so Dritschel (Reference Dritschel1995) invoked the principle of ‘adiabatic steadiness’ which requires that vortices evolve through a sequence of near-equilibrium states between inelastic interactions. This means that after one inelastic interaction the two vortices in question will return to a coherent quasi-steady state, in which the surrounding vortices modify the shape of a given vortex much slower than the intrinsic frequency of the vortex. The observed results in the present study have some discrepancies with the theoretical prediction of Dritschel (Reference Dritschel1995). In the present study, the filamentation is drawn from the smaller and weaker secondary vortex instead of the larger vortex as in the study of Dritschel (Reference Dritschel1995), and the present interaction is much fiercer, directly resulting in the breakdown of the secondary vortex. This is firstly because in the present study the vortices are unequal in strength. Secondly the upper wake vortex is not that far away, which can cause non-negligible strain to the secondary vortex and promotes its filamentation. Thirdly the vortices are in a turbulent separation flow which prohibits the existence of an equilibrium state after the inelastic interaction. Therefore, the vortex interaction in the present experiment seems to be more violent and complicated. The vortex interaction and final breakdown can increase the turbulence level, which is observed at the lower side of the cylinder (see figure 4 d). It should be pointed out that at this gap ratio, there is also the possibility of the secondary vortex being pushed down and breaking down close to the wall.

At $G/D=2.0$ in figure 15, the passing wake vortices can still induce secondary vortices periodically, but there is no longer direct interaction between the lower wake vortex and the secondary vortex because of the relatively large distance between them. It can be observed that despite of the transverse expansion of the wake, the lower wake vortex does not make contact with the vortex in the boundary layer up to $x/D=10$ . It should be noted that the visualized LCS are very similar to what has been shown by FTLE in He et al. (Reference He, Pan, Feng, Gao and Wang2016). However, in the present study only a small number of virtual tracers are used to reveal these attracting LCS, while the FTLE method requires tracers be released all over the field of view and takes significantly more computation time. As $G/D$ further increases, no secondary vortex can be observed in the boundary layer and the wake perturbations trigger the boundary layer transition in the form of a weak adverse pressure gradient.

Finally, we discuss the reason for the increase of $St$ at low gap ratios. It seems that the variation of $St$ is closely related to the interactions among the shear layers and the following vortices. Take the case of $G/D=0.5$ for example, the measured dominant frequency is the shedding frequency of the vortices, which is the same for those three vortices. In our view, the lower shear layer and the separated shear layer comprises the two sides of the gap flow, which can also be considered as a jet. At $G/D=0.5$ the jet flow is deflected away from the wall and is also strong enough to interact with the upper shear layer, the latter then provides strong disturbances that facilitates the destabilization of the two shear layers in the jet flow, which results in vortex shedding. On the other hand, the destabilized jet flow also has feedback effects on the upper shear layer, resulting in the latter shedding a vortex at the same frequency. So there are some coupling effects among the shear layers, and this coupled frequency is higher than the natural vortex shedding frequency determined by the two shear layers of a stand-alone circular cylinder, possibly due to stronger interactions in the former.

According to the above discussions, the deflection of the gap (jet) flow away from the wall and the following interaction with the upper shear layer are crucial for the increase of $St$ . This point is supported by the evidence that in the cases of $G/D=0.25$ or 0 where the gap flow is very weak or completely suppressed, the dominant frequency is no longer present on the spectra of velocity traces in the upper shear layer (see figure 5) and the following vortex shedding becomes erratic. It is also consistent with previous researches: Price et al. (Reference Price, Sumner, Smith, Leong and Paidoussis2002) and Sarkar & Sarkar (Reference Sarkar and Sarkar2010) who found the variation of $St$ at low Reynolds number ( $Re_{D}$ of a few thousands) also observed the deflection of gap flow towards the upper shear layer, while this deflection was not evident in the smoke visualization of Bearman & Zdravkovich (Reference Bearman and Zdravkovich1978) and in the vorticity plot of Wang & Tan (Reference Wang and Tan2008) who reported constant $St$ at high Reynolds number ( $Re_{D}>10^{4}$ ).