3.1. Flow visualisation

Extensive flow visualisation study was conducted using planar LIF technique and all the cases in the parameter space were considered. The videos obtained were analysed visually and many qualitative features are observed and described in this section. The observations are categorised based on the four forcing parameters as follows.

3.1.1. Effect of spacing between the cylinders

Figure 5 shows the dependence of the wake structure on spacing while keeping the forcing parameters fixed at $FR=0.8$, $\theta _0 = {\rm \pi}$ and antiphase forcing. The wake structure of the single cylinder is basically different from the regular von Kármán vortex street (observed for a stationary cylinder at $Re = 150$) in terms of the vortex size, the inter-vortex spacing, etc. These features are characteristic of a rotationally oscillating cylinder as discussed by Kumar et al. (Reference Kumar, Lopez, Probst, Francisco, Askari and Yang2013). The present discussion may be perceived as a description of morphological changes occurring in the wake structure due to proximity of the second cylinder at various spacings. The primary effect is observed at the inner shear layers between the cylinders. At $T/D = 1.4$, the vortex pair formed by the inner shear layers is forced into the wake downstream and it gets engulfed into one of the two outer shear layer vortices alternately. The wake structure repeats after every two cycles of oscillation. The interaction between the vortices is such that the outer vortex of the first cylinder interacts with the inner vortex pair to temporarily form a P+S unit while the other outer vortex advects as a single vortex. Subsequently, in the next cycle, the outer vortex of the second cylinder engulfs the inner vortex pair but also interacts further with the previously formed unit in such a way that the repetitive structure essentially forms a $\tfrac {1}{2}$(P+2S) structure based on the terminology of Williamson & Roshko (Reference Williamson and Roshko1988). Increasing the spacing to $T/D = 1.8$ creates sufficient conditions to cause symmetry in the wake structure. The inner and outer shear layers respectively form counter rotating vortex pairs that advect downstream. The mode shape changes to the 2P mode. As the spacing is still relatively low, the inner vortices are restrained in the gap region periodically and subsequently pair with the outer vortices causing an increase in the wake width. Increasing the spacing to $T/D = 2.5$ reduces the proximity effect to a considerable extent. The inner vortices are relatively less restrained in the gap region due to the increased spacing. Once formed, the inner vortex pair interacts with the outer vortices in such a way that the relative angular position of the vortices with respect to the wake centreline reduces, in other words the vortex pair alignment angle reduces (shown in the figure with white markings). With further increase in $T/D$ to 4.0, the vortex pairs are more closely spaced and are relatively stronger which is also discussed later in the section on circulation. It can also be observed that the centres of the inner vortices at the end of each cycle gets closer to the cylinders as the spacing is increased and also the vortex pair alignment angle reduces further. Increasing the spacing to a much higher value of $T/D = 7.5$, it is observed that the two vortex streets appear to be almost independent. The two vortex streets are seen to be mirror images of each other and on comparing the wake structure with that of the single cylinder, it can be observed that the vortices undergo relatively lesser diffusion in the far wake suggesting some wake interference still occurring between the two streets.

Figure 5. Effect of spacing on the wake structure at $FR=0.8$, $\theta _0 = {\rm \pi}$ with antiphase forcing. The number on each frame represents the spacing $T/D$ and the white marking shows the vortex pair alignment angle.

3.1.2. Effect of phase of oscillation

Since the roll up of shear layers from the surface of the cylinders to form vortices and the subsequent vortex dynamics dictates the wake structure, the phase difference between the two oscillating cylinders is observed to be of crucial importance. If the phase difference is zero, which is referred to as the in-phase condition, vortices are formed on the same sides of the two cylinders and the opposite is true for the antiphase mode. Another important factor affecting the wake structure is the sign of the vortices, the vortices shed in the in-phase mode are of the same sign and those shed in the antiphase mode are of the opposite sign. Figure 6 shows the effect of phase difference on the wake structure at two different spacing values of $T/D = 2.5$ and 4.0.

Figure 6. Effect of oscillation phase on the wake structure at $FR = 0.8$ and $\theta _0 = {\rm \pi}$: (a) $T/D = 2.5$ in-phase, (b) $T/D = 2.5$ antiphase, (c) $T/D = 4.0$ in-phase and (d) $T/D = 4.0$ antiphase.

In the antiphase mode, a pair of vortices is shed from the outer shear layers and a pair from the inner shear layers, alternately, in every half cycle. Since these vortices are of similar strength and opposite sign, they advect downstream parallel to the wake centreline. The wake structure, as seen in figure 6(b,d), may be called the 2P-a mode where ‘a’ refers to the antiphase condition and 2P stands for two pairs of vortices in one cycle of oscillation. The mechanism of vortex formation in the antiphase mode remains similar at $T/D = 2.5$ and 4.0; however, the size and the strength of the vortices and the spacing between them is different.

In the case of in-phase oscillation, the nature of formation and detachment of the vortices is different for $T/D = 2.5$ and 4.0. At $T/D = 2.5$, in the first half-cycle of oscillation, there is a noticeable delay observed between the detachment of the inner and outer vortices. Further, in the second half-cycle, the inner vortex of the first cylinder gets subjected to the induced motion of the second cylinder which forces it to approach and coalesce with the outer vortex of the second cylinder. Simultaneously, a part of the inner vortex of the second cylinder also gets carried away outwards intruding the coalescence and essentially forming a counter-rotating pair of vortices of unequal strengths. This process repeats for the other side of the wake in the next cycle leading to a pattern that can be called the 2P-i mode where ‘i’ refers to in-phase. The net induced velocity in the region between these pairs of vortices is in the upstream direction leading to formation of a zone of low velocity along the wake centreline which is evident from figure 6(a).

At $T/D = 4.0$, a completely different wake structure is observed in the in-phase mode (see figure 6c). Since the two consecutive vortices shed from each of the cylinders are of opposite sign, the induced velocity is in a direction perpendicular to the wake centreline. As a result, the fluid between any two subsequent vortices is displaced towards the region between the two vortices of the other cylinder leading to a structure that can be called 2S+2S. This structure, however, dissipates in the far wake as these vortices tend to coalesce at a point that depends on the forcing amplitude. The coalescence occurs because of the same sign of the subsequent vortices unlike the case in antiphase oscillation.

3.1.3. Effect of forcing frequency

Figure 7 shows the effect of forcing frequency on the wake structure at $T/D = 4.0$, an amplitude of ${\rm \pi}$ radians and antiphase forcing. The wake structure at $FR = 0$ is identified as the coupled vortex wake structure and at small values of forcing frequency like $FR = 0.25$, the wake structure is almost similar to the stationary case. At $FR = 0.5$, an interesting type of lock-in is observed where the roll up of vortices from both the sides of the cylinders is one-third part due to destructive interference and two-thirds due to constructive interference. This observation was carefully made from the flow visualisation video graph after repeated cycles of oscillation. The pattern shown in figure 7 (the frame corresponding to $FR=0.5$) repeats after every one and a half cycle of cylinder oscillation which can be called the 2/3(2P+2S) structure. Increasing the $FR$ to 0.8 leads to a very organised and symmetric wake structure which stems from complete constructive interference occurring at the cylinder surfaces. The inner and outer shear layer vortices are of similar strength (observed in the phase-averaged measurements and discussed in the section on circulation) in this case due to which the vortex pairs advect with the same velocity downstream. The vortices tend to slightly drift sideways and the inclination increases as they advect downstream leading to an increase in the wake width. As the vortices advect downstream, the inner cores of the vortices leave the plane of visualisation due to three-dimensionality forming a pyramidal structure which was clearly observed during the experiments. At $FR = 1.0$, an interesting development occurs in the wake structure which is the change in mode shape from single row to the double row mode. The spacing between vortices reduces and so does the wake width. At a certain distance downstream, the vortices tend to merge and get smeared out resulting in randomness and perhaps enhanced mixing in the far wake downstream. As $FR$ is increased to 1.2 and 1.5, the wake width reduces further and the distance between the two rows of vortices reduces. The far wake in these two cases slightly resembles the antiphase shedding mode of the vortex wake. On increasing $FR$ to 2.5, small-scale vortices are produced and the first shed vortex moves closer to the cylinders. Four rows of vortices can be observed in the near wake. These small-scale vortices coalesce and form a large vortex as observed earlier. Further, these vortices advect downstream in a manner similar to the in-phase mode of the vortex wake. As $FR$ is further increased to 4.0 and 5.0 the locked-on vortices are of even smaller size and the lock-in length decreases as the frequency is increased. For $FR = 4.0$, the far wake formed after the merger is predominantly in the antiphase mode with some instances of in-phase mode in between. At $FR = 5.0$, the far wake vortex street switches between the in-phase and antiphase modes with an almost equal probability for both the modes to occur.

Figure 7. Effect of forcing frequency on the wake structure at $T/D = 4.0$, $\theta _0 = {\rm \pi}$ with antiphase forcing. The number on each frame represents the frequency ratio $FR$.

3.1.4. Effect of forcing amplitude

The oscillation amplitude of the cylinders is varied from ${\rm \pi} /8$ to ${\rm \pi}$ radians. The effect of forcing amplitude and forcing frequency on the wake structure is observed to be coupled in nature since the mechanism of vorticity flux from the cylinder surfaces is influenced by both the parameters simultaneously. For example, the type of interference that occurs at $FR = 1.2$ for low forcing amplitudes is similar to that occurring at $FR = 0.8$ for higher amplitudes. In the present section, a high value of forcing frequency is hence chosen so that the effect of forcing amplitude could be observed independently.

Figure 8 shows the effect of forcing amplitude on the wake structure at $T/D = 4.0$, $FR = 5.0$ and in the antiphase oscillation mode. The high forcing frequency considered in the present discussion is always associated with presence of the locked-on small-scale vortices in the near wake. The difference in the wake structure, however, is due to the forcing amplitude. At low amplitude of ${\rm \pi} /8$, the small-scale vortices from each of the cylinders amalgamate to form larger vortices that advect downstream in a manner similar to the unforced coupled vortex wake of two stationary cylinders which is mostly observed to be in the antiphase mode. On increasing the amplitude to ${\rm \pi} /4$, the extent of the small-scale vortices is observed to increase slightly. In other words, the lock-in length slightly increases. On further increase in the forcing amplitude to ${\rm \pi} /2$, a significant increase in the lock-in length and a change in mode shape is observed. The two rows of vortices can be seen to extend to a much larger distance downstream. The nature of this transition to elongated shape is possibly due to a secondary instability developing at higher amplitudes as also indicated by Kumar et al. (Reference Kumar, Lopez, Probst, Francisco, Askari and Yang2013). Further increase in the amplitude to 3${\rm \pi} /4$ and ${\rm \pi}$ leads to shortening of the lock-in length. This retreat of the elongated structures in the wake with further increase in forcing amplitude (see figure 8 for $\theta _0$ = 3${\rm \pi} /4$ and ${\rm \pi}$) is found to be similar, in principle, to the response of the wake structure of an elliptic cylinder on increasing the Reynolds number as discussed by Hourigan et al. (Reference Hourigan, Thompson, Sheard, Ryan, Leontini and Johnson2007). The interaction between the individual vortex wakes beyond the point of coalescence is very minimal at forcing amplitudes of ${\rm \pi} /2$ and $3{\rm \pi} /4$ but appears to be more enhanced in the higher amplitude case of ${\rm \pi}$.

Figure 8. Effect of forcing amplitude on the wake structure at $T/D = 4.0$, $FR = 5.0$ and antiphase forcing. The number on each frame represents the oscillation amplitude.

3.2. Streamwise mean velocity profiles

To study the effect of forcing on the velocity field in the wake, time averaged (over 200 frames) streamwise mean velocity profiles are plotted and shown in figure 9. The streamwise location is fixed at $x/D = 4.0$. The ${U}_{mean}$ velocity profile for $T/D = 1.4$ has a smaller double-dip structure compared with all the other profiles and is also slightly asymmetric in nature (see figure 9a). This is due to its different mode shape, as also seen in figure 5, where due to the lower spacing, the gap flow is not large enough to cause a reduction in the velocity deficit along the wake centreline making it similar to a single bluff body. At $T/D = 1.8$, the velocity profile switches to the double-dip shape since there is a considerable flow component through the gap between the cylinders. It is further observed that maximum deficit in ${U}_{mean}$ occurs for the case of $T/D = 2.5$ and at the same time, the flow acceleration along the wake centreline is also highest for the case of $T/D = 2.5$. At higher values of $T/D = 4.0$ and 7.5, we observe that the velocity deficit behind each of the cylinders reduces and the profiles appears to be broader in shape. The reduction in deficit is due to a decrease in the net induced velocity between the counter rotating vortices in each of the rows which is in turn due to the decrease in the vortex pair alignment angle as discussed in § 3.1. At $T/D = 7.5$, the deficit reduces further and the flow velocity attains the free stream value along the wake centreline. Figure 9(b) shows the effect of oscillation phase on the streamwise mean velocity profiles. The profile at $T/D = 2.5$ displays a single dip with a large deficit for the in-phase case due to the presence of stagnation zone induced by vortices as seen in figure 6(a). The antiphase case shows a typical double-dip profile as the flow is relatively more accelerated along the mid-section due to counter-rotating vortices. The profile at $T/D = 4.0$, for the in-phase case, is observed to be significantly different since the wake displays a different mode shape (see figure 6c) where only a small deficit is observed due to relatively higher advection velocity of the vortices (observed during the experiments).

Figure 9. Effect of the forcing parameters on ${U}_{mean}$ profiles at $x/D = 4.0$, (a) spacing ($\theta _0 = {\rm \pi}$, $FR = 0.8$ and antiphase forcing), (b) phase ($\theta _0 = {\rm \pi}$, $FR = 0.8$), (c) frequency ($\theta _0 = {\rm \pi}$, $T/D = 4$ and antiphase forcing) and (d) amplitude ($T/D = 4$, $FR = 5$ and antiphase forcing).

Figure 9(c) shows the variation of ${U}_{mean}$ profiles with frequency. At $FR = 0.5$, the time-averaged velocity profile appears similar to that of the stationary case even though the instantaneous wake structure is entirely different. As the $FR$ is increased to 1.0, which is the resonant case, a drastic increase in the velocity deficit is observed. The effect of rotational oscillation of the cylinders is maximum at this forcing frequency, in other words there exists complete constructive interference between the cylinder motion and the evolving vortices. It may be effectively seen as flow past an obstacle with increased level of bluffness, resulting in lower base pressure in the near wake. With further increase in $FR$ to 1.5 and 2.5, the velocity deficit starts to reduce since the interference effect on the vortex formation is only partial, that is, the cylinder surface motion and the rolling up of shear layers are partly corotating and partly counter-rotating which leads to occurrence of both constructive and destructive interference. At $FR = 5.0$, the presence of small scale vortices and a region of low velocity along the centreline in the near wake once again leads to an increase in the velocity deficit. Figure 9(d) shows the effect of amplitude. The profiles for $\theta _0 = {\rm \pi}/8$ and ${\rm \pi} /4$ appear similar to the case of two stationary cylinders due to the coupled vortex wake structure. As the amplitude is increased to ${\rm \pi} /2$, a region of very low velocity is observed to exist along the cylinder centrelines which occurs due to the change in mode shape occurring at $\theta _0 = {\rm \pi}/2$ as discussed previously. The width of the profile is also low due to the narrower wake width present at $FR = 5.0$. The behaviour of the streamwise velocity profiles at higher amplitudes is similar to that of ${\rm \pi} /2$ at the considered location of $x/D = 4.0$.

3.3. Cross-stream mean velocity profiles

Figure 10 shows the variation of ${V}_{mean,p}$ (where the subscript $p$ refers to the phase-locked condition) across the wake at $x/D = 4.0$ which was obtained from the phase-averaged (over 100 frames) PIV data at the same forcing conditions mentioned in the previous section while the phase was locked at maximum amplitude position of the cylinders. It can be noted that the magnitude of ${V}_{mean,p}$ is greatly amplified by the presence of second cylinder when compared with the single-cylinder case. With an increase in spacing between the cylinders, the lateral deflection of the fluid increases considerably (see figure 10a). The velocity induced in the cross-stream direction by the vortex pairs increases significantly when the vortices are more aligned along the cylinder centreline. In addition to this, the higher vortex strength at higher $T/D$ values also contributes to the cross-stream motion of the fluid. The case corresponding to the single cylinder displays a single-sided deflection because of the phase-locked condition. Figure 10(b) shows the variation of ${V}_{mean,p}$ with phase. For the antiphase forcing condition, one can observe that, at both $T/D = 2.5$ and 4.0, the variation of ${V}_{mean,p}$ is symmetrical in nature with both positive and negative values corresponding to the instantaneous position of the vortices. For the in-phase forcing condition, a significant deflection of the flow can be observed in the cross-stream direction which is due to the same sign of the vortices and the considered angular position of the cylinders. One can also observe the unequal nature of the profile at $T/D = 2.5$ since vortex pairs from each of the cylinders advect alternately in the downstream direction (see figure 6a) unlike the case of $T/D = 4.0$ where the vortices from both the cylinders advect simultaneously (see figure 6c).

Figure 10. Effect of the forcing parameters on ${V}_{mean,p}$ profiles at $x/D = 4.0$: (a) spacing ($\theta _0 = {\rm \pi}$, $FR = 0.8$ and antiphase forcing), (b) phase ($\theta _0 = {\rm \pi}$, $FR = 0.8$), (c) frequency ($\theta _0 = {\rm \pi}$, $T/D = 4$ and antiphase forcing) and (d) amplitude ($T/D = 4$, $FR = 5$ and antiphase forcing).

Figure 10(c) shows the effect of frequency. One can observe that there is maximum amount of flow deflection at $FR = 0.8$ which is due to the presence of stronger vortex pairs. The variation at $FR = 1.0$ and 1.5 is also considerable but could not be captured completely in the phase-locked condition at the chosen downstream location. At higher forcing frequencies, minimum variation in ${V}_{mean,p}$ is observed since the vortices are relatively weaker. Figure 10(d) shows the effect of amplitude. The variation of ${V}_{mean,p}$ is considerable at lower amplitudes of ${\rm \pi} /8$ and ${\rm \pi} /4$ due to higher amounts of vortex deflection in the cross-stream direction. One can observe that, at the chosen location of $x/D = 4.0$, the small-scale vortices coalesce to form a bigger vortex which further gets swayed in cross-stream direction to form the coupled vortex wake structure. At forcing amplitudes of ${\rm \pi} /2$ and $3{\rm \pi} /4$, there is minimum amount of cross-stream entrainment and, hence, negligible variation of ${V}_{mean,p}$ at the downstream location considered. At much higher amplitude of ${\rm \pi}$, the shortening of the lock-in length results in formation of large scale vortices at a relatively smaller distance downstream which further leads to an increase in ${V}_{mean,p}$ variation across the wake.

3.4. Streamwise mean velocity recovery

The behaviour of the near-wake region for flows around bluff bodies is crucial in determining characteristics such as recirculation length, wake width, cross-stream entrainment and eventually the drag force. A reduction in recirculation length corresponds to faster recovery of mean velocity along the wake centreline and further contributing to drag reduction (Dutta, Panigrahi & Muralidhar Reference Dutta, Panigrahi and Muralidhar2008). This behaviour is also reported in the case of streamwise oscillating cylinder (Dutta, Panigrahi & Muralidhar Reference Dutta, Panigrahi and Muralidhar2007; Konstantinidis, Balabani & Yianneskis Reference Konstantinidis, Balabani and Yianneskis2003). In the present forcing conditions, however, the surface acceleration of the cylinder affects the recirculation region in a different way. At resonant frequencies, the recirculation region (based on the peak value of ${u}_{rms}$ along the centreline of the cylinder) reduces and at higher frequencies it spans the entire lock-in length of the vortex wake. It is later discussed in § 3.9 that drag attains a maximum value at resonant frequencies and a minimum value at higher frequencies. It may hence be noted that recirculation region and the corresponding recovery of mean streamwise velocity alone does not influence drag, instead other factors such as the extent of velocity fluctuations and also the wake width together with the size of recirculation region influence the drag force in the case of rotationally oscillating cylinder. The nature of velocity recovery in the downstream direction in the present forcing scenario demonstrates the possibility of vortex coalescence and also indicates the presence of low-velocity regions in the wake.

In the present section the recovery of ${U}_{mean}$ and its variation with forcing parameters along the centreline of one of the cylinders is discussed as shown in figure 11. The values of forcing parameters are kept consistent with those in § 3.2. Figure 11(a) shows the effect of spacing. At $T/D = 7.5$, the two wakes are essentially independent and the mean velocity recovery is that of an equivalent single rotationally oscillating cylinder. As the spacing is reduced to $T/D = 4.0$, the recovery rate is relatively lower in the near wake and then steadily increases downstream. With further reduction in spacing to lower values, the trend is observed to be opposite where initially there is a delay observed in recovery followed by a substantial increase downstream until it reaches the asymptotic value. On comparing the two extreme cases of spacing, under the present condition of forcing, it may be deduced that as the cylinders are brought closer, the vortex pair alignment angle increases along with an increase in the lateral distance between the two vortex pairs which causes reasonable acceleration of the fluid in the far wake towards the asymptotic value. In the case of a single cylinder or $T/D = 7.5$, the fluid particles in the immediate downstream region of the cylinders are constantly swayed across without any proximity effects of the second cylinder and, hence, always contain a considerable component of streamwise velocity compared with the case of a stationary cylinder where there would be a stagnation region. This possibly explains the sudden rise in ${U}_{mean}$ value for $T/D = 7.5$ and for the single cylinder.

Figure 11. Effect of the forcing parameters on ${U}_{mean}$ recovery along the centreline of the top cylinder: (a) spacing ($\theta _0 = {\rm \pi}$, $FR = 0.8$ and antiphase forcing), (b) phase ($\theta _0 = {\rm \pi}$, $FR = 0.8$), (c) frequency ($\theta _0 = {\rm \pi}$, $T/D = 4$ and antiphase forcing) and (d) amplitude ($T/D = 4$, $FR = 5$ and antiphase forcing).

Figure 11(b) shows the effect of oscillation phase. There is a significant difference observed in the recovery trend of ${U}_{mean}$ in the in-phase and antiphase cases for $T/D = 2.5$. The wider structure of the wake present in the case of in-phase forcing leads to a poorer recovery whereas a steep increase is observed in the antiphase case indicating a faster recovery and lesser deficit in velocity. The recovery of ${U}_{mean}$ in the in-phase forcing case of $T/D = 4.0$ shows a completely opposite behaviour to that of $T/D = 2.5$. There is a large increase in the mean velocity value in the immediate downstream ($x/D = 2$) region of the wake and it recovers completely around $x/D = 7$ and then again starts to decrease gradually further downstream. The decrement in the downstream part is attributed to the merger of vortices as also seen in figure 6(c). An important observation that can be made about the difference in the recovery trend at these two spacings is the proximity interference effect which is more dominant in the $T/D = 2.5$ case where the inner vortices of each of the cylinders are highly influenced by the motion of the other cylinder. At higher value of $T/D = 4.0$, the vortices detach from the cylinders rather independently and later interact with other vortices in the wake.

Figure 11(c) shows the effect of forcing frequency. At $FR = 1.0$, it can be observed that the mean velocity remains low in the near wake which is due to the resonance frequency state resulting in stronger vortices and an extended region of momentum deficit. This effect starts to reduce at moderate values of forcing frequencies such as $FR = 1.5$ and 2.5 and also the far-wake structure becomes relatively more ordered leading to an increase in the recovery of ${U}_{mean}$. One can expect a high asymptotic value of ${U}_{mean}$ at these forcing frequencies. At $FR = 5.0$, there is an initial delay in the recovery of ${U}_{mean}$ due to the stagnation region formed and later a high recovery rate is observed beyond the lock-in length.

Figure 11(d) shows the effect of forcing amplitude. The recovery trend of ${U}_{mean}$ observed at lower amplitudes is similar to that of the unforced wake whereas at higher amplitudes the narrow and elongated near-wake structure delays the initial recovery. It is also observed that, at higher forcing amplitudes beyond the lock-in length, the recovery is enhanced due to the coupled vortex wake structure present in the far wake. An interesting observation from the plot is the asymptotic limit of the ${U}_{mean}$ that is attained at different forcing amplitudes. At lower amplitudes, even though the initial recovery is high, the maximum value attained is around 75 % of the ${U}_{\infty }$ whereas at $\theta _0 = {\rm \pi}/2$ and $3{\rm \pi} /4$, the asymptotic value reached is relatively higher which leads to a much lower momentum deficit in the wake. This behaviour is an indicative of the lower values of drag observed at these forcing amplitudes.

3.5. Peak velocity deficit

The magnitude of the peak deficit in ${U}_{mean}$ at a point in the near wake of the cylinder is indicative of the influence of the cylinder bluntness and the momentum of the wake fluid. It may be noted that the global wake structure in the present forcing conditions depends on factors such as vortex coalescence, narrowing of the wake width, small-scale vortices and velocity fluctuations. In the present section, however, only the mean velocity defect and its variation with forcing is studied (see figure 12). The streamwise location is fixed at $x/D = 4.0$. For the single stationary cylinder, in the present setting of $Re = 150$, the peak value of the deficit is observed to be 0.3, in other words, the maximum deficit is 30 % and this value increases significantly with forcing as seen in figure 12(a). It can be noted that the peak deficit is independent of forcing amplitude for $FR<1.0$ and a similar behaviour can be seen for lower amplitudes in figure 12(b) which indicates that the mean velocity deficit is more prominent for amplitudes of ${\rm \pi} /2$ and higher and for $FR>1.0$. The drastic increase in the peak velocity deficit at $FR = 1.0$ is attributed to the change in mode shape from a single row to the double row structure in the wake where the two rows of vortices induce a velocity in the upstream direction along the cylinder centreline. This effect prevails even for higher values of amplitudes and $FR$ owing to the presence of the double row mode.

Figure 12. Effect of forcing on the peak velocity deficit at $x/D = 4$: (a,b) single cylinder, (c,d) two cylinders in in-phase and antiphase, respectively, at $\theta _0 = {\rm \pi}$.

Figure 12(c,d) show the variation of peak deficit with $FR$ at a fixed amplitude of ${\rm \pi}$ for different spacing values. The values are extracted only from one of the two cylinder ${U}_{mean}$ profiles. It can be observed that if the forcing amplitude is relatively higher, the value of the peak deficit follows a scaling behaviour for $FR>1.0$ and the trend remains the same for both in-phase and antiphase forcing for all values of spacing. When $FR<1.0$, the presence of second cylinder affects the local velocity field differently for in-phase and antiphase conditions and the nature of vortex interaction in the wake for different forcing conditions essentially influences the ${U}_{mean}$ value. For example, at $T/D = 4.0$, the velocity deficit at $FR = 1.0$ is very low in the in-phase forcing compared with the value in antiphase forcing. Similarly, at $T/D = 1.4$, the peak deficit value at $FR = 0.5$ is much lower in the in-phase forcing than in the antiphase forcing and the opposite behaviour can be seen at $FR = 1.0$. Another observation that can be made from the $Y$-axis values in figure 12(c,d) is that the peak deficit is higher for lower $T/D$ when the cylinders are stationary.

To further investigate the variation of the peak deficit with forcing, scaling analysis using linear least squares method was carried out. The peak deficit scaled with forcing amplitude shows a reasonable fit to the data when considered in two parts, one for $0.25< FR<0.8$ and the other for $1.2< FR<5.0$ (see figure 13b). The value of the scaling exponent $n$ is 0.5 as shown in the inset in figure 13(b). For the case of scaling with $FR$, such a scaling could not be obtained. However, linear fits are done to approximate the data as shown in figure 13(a). For the case of two cylinders in figure 12(c,d), the peak deficit could closely be approximated by fourth-order polynomial fit for the data beyond $FR$ 1.0 as shown with the dashed curves in the figure.

Figure 13. Variation of scaled peak deficit for the single cylinder at $x/D = 4$: (a) with $\theta _0$ and (b) with $FR$. The value of scaling exponent, $n$ is 0.5 for $1.2< FR<5.0$ and 0.1 for $0.25< FR<0.8$. The inset shows the residual variation with $n$.

3.6. Wake width

In this section the wake width and its variation with forcing is discussed. The wake width $w$ in the present study is defined as the cross-stream separation of $y$ location where the ${U}_{mean}$ attains the ${U}_{\infty }$ value. This definition is also used in Dutta et al. (Reference Dutta, Panigrahi and Muralidhar2008). The downstream location where the wake width is measured is $x/D = 20$. The objective is to study the wake behaviour at a distance relatively far from the cylinder so that the factors such as vortex coalescence and lock-in length can also be included. Figure 14 shows the variation at different values of $T/D$. It can be noted that the wake width for a single cylinder is maximum at $FR = 0.5$ and minimum at $FR = 2.5$ and 4.0 whereas the maxima and minima for the two-cylinder wakes is seen to be also affected by the phase difference and the spacing. The single-cylinder wake at $FR = 0.5$ undergoes transition from an unforced-like wake to the locked-on wake resulting in considerable transverse deflection of the fluid and as $FR$ reaches 1.0, the wake mode switches to the locked-on double-row mode where the width reduces but still remains higher than the stationary value. As $FR$ increases further, the two rows of vortices move closer towards the wake centreline leading to a reduction in the wake width. At much higher values of $FR$, the effect of lock-in starts to reduce and the wake behaves as an unforced wake.

Figure 14. Variation of wake width with $FR$ at different $T/D$. Wake width is calculated at $x/D = 20$, and only $\theta _0 = {\rm \pi}$ is considered.

At $T/D = 1.4$, the wake width attains a maximum value at $FR = 0.5$ for the in-phase forcing. This is attributed to its distinctive wake structure (discussed further in § 4.1.3) where there is large amount of fluid displacement across the wake. On the other hand, the antiphase forcing results in a narrower wake when compared with a stationary cylinder. At higher forcing frequencies, there is no noticeable difference between the single- and two-cylinder wake widths. At $T/D = 1.8$, the variation in the wake width is similar to the $T/D$ 1.4 case except for two major differences. At $FR = 4.0$ and 5.0, the increased gap between the cylinders avoids the formation of stagnation zone which, in turn, leads to a narrower wake in the downstream portion. Furthermore, the presence of 2P-a mode along with the high value of vortex pair alignment angle present in the antiphase case at $FR = 0.8$ gives rise to high wake width as also discussed in § 3.1.1.

At $T/D = 2.5$, there is a noteworthy difference between the in-phase and the antiphase forcing, wherein the in-phase nature of vortex shedding apparently creates more diffusion in the far wake and, as such, large-scale structures are not formed leading to low values of wake width ($FR = 1.5$ and 2.5) and in the antiphase forcing, coalescence of vortices in the far wake gives rise to an increase in wake width. One can also note that at $FR = 0.8$, the wake width is much higher in the case of in-phase forcing due to larger deflection of the fluid laterally. At $T/D = 4.0$, the trend is similar to that of the single cylinder beyond $FR = 1.0$, but for $FR<1.0$, the in-phase forcing case has a slightly larger wake width.

3.7. Fluctuation intensity

Variation in the fluctuation levels indicate a possible enhancement or diminution in the amount of mixing occurring in the wake. The following expression is employed to study the fluctuation levels in the wake where the root-mean-squared value of the fluctuating components is normalised with ${U}_{\infty }$:

(3.1)\begin{equation} I = \frac{\sqrt{(u_{rms}^2 + v_{rms}^2)}}{U_{\infty}}. \end{equation}

In this section, only the effect of frequency is discussed while the other forcing parameters are fixed. Figure 15 shows the time-averaged fluctuation intensity fields for different values of $FR$. The surface acceleration of the cylinders essentially results in injection of additional vorticity into the evolving vortices. Figure 15(b) shows a moderate increase in fluctuation levels when the cylinders are forced to oscillate at $FR = 0.5$. At the resonant forcing frequency, one can observe that the wake contains peak fluctuation intensity in most of the downstream region. Beyond $FR = 1.0$, the cylinders and the evolving vortices, due to higher oscillation frequency, undergo destructive interference and the strength of the vortices reduces in the wake resulting in relatively low levels of fluctuation as seen in figure 15(e). At much higher value of $FR = 5.0$, the lock-in length, which is the effective region of influence of the cylinder oscillation in the wake, is observed to decrease which further leads to coalescence of vortices forming large-scale rotating structures and again causing the fluctuation levels to increase downstream (see figure 15f).

Figure 15. Effect of forcing frequency on the fluctuation intensity $I$ distribution at $\theta _0 = {\rm \pi}$, $T/D = 4.0$ and antiphase forcing: (a) stationary, (b) $FR = 0.5$, (c) $FR = 1.0$, (d) $FR = 1.5$, (e) $FR = 2.5$ and ( f) $FR = 5.0$.

3.8. Circulation and vorticity field

The characteristics of the wake vortices in the present investigation are studied by phase averaging the PIV data, that is, obtaining PIV data for a particular value of the angular position of the cylinders and then averaging over multiple cycles of oscillation. In the present investigation, the position of maximum angular displacement of the cylinders was selected at each value of forcing frequency and amplitude for both in-phase and antiphase forcing conditions. The vorticity of the nearest shed vortex from both the outer and inner shear layers of one of the cylinders is first obtained and then a threshold cut-off of 5 % of the peak vorticity is applied to determine an area of positive or negative vorticity followed by an area integral performed to obtain the value of circulation. It was observed that upon changing the cut-off to 10 %, the circulation values only changed by a small amount ($\le 6\,\%$). The circulation values are non-dimensionalised using the free stream velocity ${U}_{\infty }$ and diameter of the cylinder $D$ i.e. ${\varGamma }/ {U}_{\infty } D$.

First, the procedure was carried out on a single rotationally oscillating cylinder and the variation of the circulation is determined as shown in figure 16. The oscillation amplitude is fixed at ${\rm \pi}$ radians. It can be observed that the peak value of circulation occurs around the resonant frequency and a decreasing trend is observed as the frequency is increased. This variation is consistent with the observations made by Kumar et al. (Reference Kumar, Lopez, Probst, Francisco, Askari and Yang2013). The reduction in the strength of the vortices is due to the effect of destructive interference between the cylinder surfaces and the evolving vortices, that is, during the vortex formation process the cylinder and the forming vortex rotate in the same direction if the forcing frequency is higher and thereby causing weakening of the vortices. On the other hand, in the resonant frequency band, the cylinder and the forming vortex rotate in the opposite direction which causes an additional influx of vorticity and an increase in the strength of the vortices. In the present investigation, the presence of an additional cylinder and its proximity effects under the forcing conditions lead to an increase or decrease in the relative strength of the vortices. The spacing between the cylinders also affects the vortices forming at the cylinder surfaces considerably. As $T/D$ is decreased, the similarity in the strength and size of the outer and inner shear layer vortices of each of the cylinders starts to reduce. Figure 16 also shows the relative behaviour of the outer and inner shear layer vortices and their dependence on spacing for both in-phase and antiphase oscillations. The $\theta _0$ and $FR$ were fixed at ${\rm \pi}$ and 1.0, respectively. At higher values of spacing, minimum variation in the circulation values is observed with oscillation phase and also no significant difference is observed between the inner and outer shear layer vortices. This can be attributed to relatively lower amount of interaction between the shed vortices at higher values of $T/D$. As the spacing is reduced, the proximity interference between the inner shear layers of the two cylinders leads to a decrease in the strength of these inner vortices as compared to the outer vortices which are exposed to the free stream on the other side. This leads to a gradient in vorticity in the outward cross-stream direction also causing an increase in the wake width.

Figure 16. Variation of circulation with $FR$ at $\theta _0 = {\rm \pi}$ for a single cylinder and the variation with spacing at $FR = 1.0$ and $\theta _0 = {\rm \pi}$. OSL and ISL denote outer and inner shear layers, respectively.

The variation of the circulation values with forcing frequency at various forcing amplitudes is shown in figure 17 for both in-phase and antiphase oscillations at $T/D = 2.5$. The insets in the figure show the forcing condition and the nearest shed vortex considered for the phase-averaged calculations. It is observed that the interaction between the inner and outer shear layer vortices of the two cylinders gives rise to a difference in their circulation values as can be seen in figure 17(a,c). When the cylinders oscillate in the antiphase manner, no significant difference is observed in the circulation values of outer and inner shear layer vortices since these vortices are symmetric about the wake centreline advecting downstream with equal velocity. However, for the in-phase oscillation case, the strength of the outer shear layer vortices is much higher than that of the inner vortices at resonant frequency and also the peak value occurs for a relatively lower value of $FR$. At higher values of $FR$, neither the amplitude nor the oscillation phase has any significant effect on the vortex strength since the near-wake structure at higher forcing frequencies is essentially the same at all amplitudes where the small-scale vortices of the locked-in region are seen to dominate. The effect of forcing amplitude on the vortex strength is observed to be of relatively lesser significance.

Figure 17. Variation of circulation with forcing frequency at different oscillation amplitudes at $T/D = 2.5$. The insets show the corresponding vortices considered: (a) inner shear layer vortex with in-phase forcing, (b) inner shear layer vortex with antiphase forcing, (c) outer shear layer vortex with in-phase forcing and (d) outer shear layer vortex with antiphase forcing.

The values of circulation beyond $FR$ 1.0 are observed to follow a decaying trend as also noted by Kumar et al. (Reference Kumar, Lopez, Probst, Francisco, Askari and Yang2013). There is no noticeable effect of the phase or amplitude on the circulation values beyond $FR$ 1.0. A power law curve fit of the form $y=ax^{b}$ is done for all the circulation values for the first detached vortex and they are also shown in figure 18. The constants $a$ and $b$ for different configurations of the cylinders are presented in table 1.

Figure 18. Power law fits of the form $y=ax^{b}$ for circulation values in different configurations for $FR\ge 1.0$ and $T/D = 2.5$. The data points represent the averaged values for various amplitudes.

Table 1. Values of the coefficients in the power law expression $y=ax^{b}$ for different configurations of the two cylinders at $T/D = 2.5$. Single-cylinder data are also listed for comparison.

Figure 19 shows the vorticity contours obtained from the phase-averaged data. The vorticity is normalised by the free stream velocity U$_\infty$ and diameter of the cylinder $D$, i.e. $\omega D/U_\infty$. The amplitude is fixed at $\theta _0 = {\rm \pi}$ and only two spacing values are considered in the present discussion. The qualitative wake structure is already discussed in the earlier sections, here the phase-averaged contours of vorticity are discussed with an objective to elucidate the changes occurring in the mode shape while the forcing frequency is varied. At $T/D = 2.5$, in the in-phase forcing condition, the wake structure undergoes transformation from the 2P-i mode (at $FR = 0.8$) to the P+S mode (at $FR = 1.2$) whereas in the antiphase case, no such mode shape change is observed. At higher frequency value of $FR = 2.5$, the wake structure follows the 2P-double row mode in both in-phase and antiphase forcing conditions. At $T/D = 4.0$, a different transition is observed at $FR = 1.0$. The 2S+2S wake structure in the in-phase case and the 2P wake structure in the antiphase case both change to the double row mode at $FR = 1.2$. However, the topological configuration for both the cases is different as shown in the insets.

Figure 19. Contours of vorticity at $\theta _0 = {\rm \pi}$ for different values of $FR$ and $T/D$. The contour levels shown do not include extreme values of the vorticity.

3.9. Determination of drag

An estimate of drag is made using the time-averaged PIV data based on the mean momentum deficit in the streamwise direction and also due to the contributions from velocity fluctuations by applying a momentum balance approach over a control volume around the cylinders (Dutta et al. Reference Dutta, Panigrahi and Muralidhar2008). Figure 20 shows the schematic of the control volume used for drag calculation. The control surfaces labelled 2 and 3 have a length $L$ and the edge containing the cylinders was considered as the free stream region with a constant ${U}_{\infty }$ value for all the calculations. The size of the domain and the free stream velocity value was kept constant for all the cases considered in order to have consistency in the results. In addition to this, an assumption prevalent in our study is that the flow is two dimensional which strictly is not true since the forcing conditions greatly alter the flow in the third dimension depending on the forcing frequency and amplitude (Kumar et al. Reference Kumar, Lopez, Probst, Francisco, Askari and Yang2013; Bhattacharyya et al. Reference Bhattacharyya, Khan, Verma, Kumar and Poddar2022). In the present study a region consisting of $22D$ downstream and $13D$ cross-stream was considered and the integrals given by the following equations were computed:

(3.2)$$\begin{gather} C_d=\frac{2}{(2D)} \int_{-\infty}^{\infty} \frac{\bar{U}}{U_{\infty}}\left(1-\frac{\bar{U}}{U_{\infty}}\right) \,{{\rm d} y}, \end{gather}$$

(3.3)$$\begin{gather}C_d=\frac{2}{(2D)} \int_{-\infty}^{\infty} \frac{\bar{U}}{U_{\infty}}\left(1-\frac{\bar{U}}{U_{\infty}}\right)\, {{\rm d} y} + \frac{2}{(2D)} \int_{-\infty}^{\infty} \frac{v^{2}_{rms}-u^{2}_{rms}}{{U_{\infty}}^2}\, {{\rm d} y}. \end{gather}$$

Figure 20. Schematic of control volume specification for drag calculation. The control surface 1 is assumed to possess a constant ${U}_{\infty }$ value.

This method of drag estimation by measurement of the velocity profiles depends upon the downstream location and ideally a distance of around $30D$ is considered adequate (Antonia & Rajagopalan Reference Antonia and Rajagopalan1990). In the present experiments we had a PIV window corresponding to $22D$ which for some of the cases gave rise to conditions where the terms in (3.2) and (3.3) failed to attain constant values with respect to the downstream distance. Keeping this in mind, a more detailed analysis was attempted by considering the contributions from various other possible sources. One such factor is the pressure distribution at the streamwise measurement location as suggested by Jin, Wu & Choi (Reference Jin, Wu and Choi2021). In order to express the pressure distribution in terms of the velocity field, the y-component of the Navier–Stokes equation was considered after neglecting the mean transverse velocity and streamwise gradient of the Reynolds stresses:

(3.4)\begin{equation} \frac{\partial{P}_{avg}}{\partial{y}} = \frac{\partial{v^{2}_{rms}}}{{\partial{y}}}. \end{equation}

Bernoulli's condition was applied at control surface 4 of the control volume and the resulting expression for pressure is given by

(3.5)\begin{equation} P_{avg}(y) = P_{\infty} + \tfrac{1}{2} \rho ({U_{\infty}}^2 - {U_0^{2}}) - \rho v^{2}_{rms} (y), \end{equation}

where $U_0$ is the value of velocity at the edge of control surface 4. Further, expressing in terms of pressure coefficient and adding to (3.3) gives rise to

(3.6) $$\begin{align} C_d&=\frac{2}{(2D)} \int_{-\infty}^{\infty} \frac{\bar{U}}{U_{\infty}}\left(1-\frac{\bar{U}}{U_{\infty}}\right) \,{{\rm d} y} + \frac{2}{(2D)} \int_{-\infty}^{\infty} \frac{v^{2}_{rms}-u^{2}_{rms}}{{U_{\infty}}^2}\, {{\rm d} y}\nonumber\\ &\quad + \frac{1}{(2D)} \int_{-\infty}^{\infty} \left(\frac{U_0^{2}}{{U_{\infty}}^2} - 1\right)\, {{\rm d} y}. \end{align}$$

A more recent study by Sunil et al. (Reference Sunil, Kumar and Poddar2022) has further investigated the effects of other additional terms such as the streamwise component of momentum flux across control surfaces 2 and 3 and variation of transverse mean velocity at control surface 4. The following equation represents the contribution of these terms in addition to those present in (3.6):

(3.7)\begin{align} &C_d=\frac{2}{(2D)} \int_{-\infty}^{\infty} \frac{\bar{U}}{U_{\infty}}\left(1-\frac{\bar{U}}{U_{\infty}}\right) \,{{\rm d} y} + \frac{2}{(2D)} \int_{-\infty}^{\infty} \frac{v^{2}_{rms}-u^{2}_{rms}}{{U_{\infty}}^2} \,{{\rm d} y}\nonumber\\ &\quad + \frac{2}{(2D)} \int_{-\infty}^{\infty} \frac{\bar{V}}{U_{\infty}}\, {{\rm d} y} + \frac{1}{(2D)} \int_{-\infty}^{\infty} \left(\frac{U_0^{2}}{{U_{\infty}}^2} - 1\right) \,{{\rm d} y} + \frac{2}{(D)} \int_{0}^{L} \frac{\overline{V_3}}{U_{\infty}}\left(1-\frac{\overline{U_2}}{U_{\infty}}\right)\, {{\rm d} x} \nonumber\\ &\quad - \frac{2}{(D)} \int_{0}^{L} \frac{\overline{V_2}}{U_{\infty}}\left(1-\frac{\overline{U_3}}{U_{\infty}}\right)\,{{\rm d} x}. \end{align}

Since there are two cylinders, the total drag force obtained is non-dimensionalised by $(2D)$ to obtain ${C}_{d}$ above as follows:

(3.8)\begin{equation} C_d= \frac{F_{drag}}{\frac{1}{2}\rho{U_{\infty}}^2 (2D)}. \end{equation}

It was noted that the drag coefficient values resulting from (3.2) and (3.3) underestimated the drag coefficient and the modified version in (3.6) significantly improved the drag estimation. A good agreement with the direct force measurement values was observed. The possible reason for this difference is the considerable amount of transverse velocity fluctuation occurring at control surface 4 and the difference in values of free stream velocity at control surfaces 1 and 4. It is further be observed that the contributions from the additional terms in (3.7) are found to be insignificant.

In the present investigation, a strain-gauge-based direct drag measurement set-up was also designed and constructed in our laboratory with an objective of determining the precise value of drag. Multiple trials of calibration were performed initially with different load cells and the low magnitude of the drag force required the measurements to be highly sensitive. The experiments were repeated more than 10 times each and the standard deviation is indicated in the presented data. To validate the data obtained from the set-up, ${C}_{d}$ values for a single stationary cylinder at various $Re$ are obtained and shown in figure 21 with the existing data from Panton (Reference Panton1984) and von Wieselsberger (Reference von Wieselsberger1921). To the best of the authors’ knowledge, there is no evidence of experimentally obtained value of ${C}_{d}$ for the case of two cylinders at $Re = 150$. Comparison is made with the numerical results obtained by Zhao (Reference Zhao2013).

Figure 21. Validation: (a) ${C}_{d}$ variation of single stationary cylinder with $Re$ and (b) ${C}_{d}$ variation of two stationary cylinders with spacing at $Re = 150$.

It can be seen in figure 21(b) that the ${C}_{d}$ value initially increases with spacing and then decreases approaching the single cylinder value. The variation follows a trend similar to that of Zhao (Reference Zhao2013). An important observation is that the drag is not directly proportional to the increase in spacing, instead, it reaches a maximum value at $T/D = 2.5$ and further approaches the value of the isolated cylinder which marks the transition between the biased bistable wake mode and the coupled vortex street mode observed for $T/D\leq 2$ and $T/D\geq 2.5$, respectively, based on the terminology of Zdravkovich (Reference Zdravkovich1987).

In order to study the effect of the forcing parameters on drag of two cylinders, first a single rotationally oscillating cylinder was considered and the drag variation with forcing was studied followed by the two-cylinder cases. Any relative enhancement or reduction in drag in the case of two cylinders is then attributed to the interaction between the two vortex streets. The oscillation amplitude was fixed at ${\rm \pi}$ radians and variation with forcing frequency is plotted as shown in figure 22. The values are normalised with the stationary cylinder value, ${C}_{d0}$. The drag coefficient as expressed by (3.6) and (3.7) is found to agree well with the direct force measurement data which are shown in solid symbols. Beyond $FR>2.5$, the direct force measurements could not be made due to constraints from the motor vibrations. It can be observed in figure 22(a) that there is a peak value at $FR = 1.0$ and a considerable diminution at higher forcing frequencies. On comparing the behaviour of the wake at resonant frequencies in terms of circulation value, the fluctuation intensity and the mean velocity deficit, a strong correlation can be observed, that is, the forcing condition for maximum drag is the same condition at which a peak in circulation as well as fluctuation intensity is observed. On the other hand, the reduction in drag at higher frequencies can be related to the minima in circulation values and fluctuation intensity, respectively. Hence, the mechanism of formation and propagation of vortices in the wake of a rotationally oscillating cylinder at a given forcing condition may either lead to a sizable enhancement or reduction in drag compared with the stationary cylinder value. An interesting observation that can be made at $FR = 5.0$ is that the ${C}_{d}$ value tends to increase, this behaviour can be related to its wake structure where the locked-in portion starts to reduce and the far wake contains larger rotating structures compared with $FR = 4.0$ where the entire wake is composed of small-scale vortices.

Figure 22. Variation of drag coefficient with $FR$ at $\theta _0 = {\rm \pi}$: (a) single cylinder; (b) two cylinders, in-phase at $T/D = 1.4$; (c) two cylinders, antiphase at $T/D = 1.4$. Here ${C}_{d0}$ refers to the value of drag of stationary cylinder(s). The different expressions for ${C}_{d}$ as represented by (3.2)–(3.7) are shown in different colours.

To study the effect of forcing on the drag of two cylinders, three values of $T/D$ were chosen and both in-phase and antiphase forcing conditions were considered as shown in figures 22(b,c) and 23. The oscillation amplitude was fixed at ${\rm \pi}$ radians and the values obtained are normalised by the drag coefficient of two stationary cylinders at the corresponding $T/D$. It can be observed that the peaks and the minima in drag are noticeably different for the three spacing values considered.

Figure 23. Variation of drag coefficient with $FR$ at $\theta _0 = {\rm \pi}$: (a) $T/D = 2.5$ in-phase; (b) $T/D = 2.5$ antiphase; (c) $T/D = 4.0$ in-phase; (d) $T/D = 4.0$ antiphase. Here ${C}_{d0}$ refers to the value of drag of stationary cylinder(s). The different expressions for ${C}_{d}$ as represented by (3.2)–(3.7) are shown in different colours.

At $T/D = 1.4$, in the antiphase forcing condition, the wake is observed to be similar to that of the two stationary cylinders at low forcing frequencies whereas in the case of in-phase forcing, the value of drag is seen to be relatively higher. There is an interesting behaviour occurring at $FR = 0.5$ (see part C of figure 25) where the contribution from mean velocity deficit and fluctuations is significantly large but the contribution due to pressure is in the opposite trend owing to the low value of $U_0$ at control surface 4. Beyond $FR = 1.0$, the condition of maximum constructive interference is observed to prevail at $FR = 1.2$ irrespective of the phase of oscillation and, hence, there is a peak observed for both the cases. Further increase in frequency to $FR = 2.5$ causes a slight reduction in drag. At much higher values of $FR = 4.0$ and 5.0, the drag force again increases which suggests that beyond the locked-in region, the resultant wake structure is more than twice as effective as two isolated cylinders since it contains amalgamated structures from the vortex streets of the two cylinders.

Figure 24. Wake mode shapes and boundaries for $T/D = 1.4$, (a) in-phase and (b) antiphase: A, near-wake biased; B, near-wake biased and locked-on; C, 2S, single bluff body wake, in-phase; D, 2S, double row; E, far-wake single vortex street; F, $\tfrac {1}{2}$(P+2S); G, $\tfrac {1}{2}$(2S), single bluff body wake, antiphase; H, 2P double row; ($\cdots$) unidentifiable/irregular wake pattern.

Figure 25. Different wake modes: A, near-wake biased; B, near-wake biased and locked-on; C, 2S single bluff body wake, in-phase; D, 2S, double row; and E, far-wake single vortex street.

Upon increasing the spacing to $T/D = 2.5$, the trend in the variation of ${C}_{d}$ in the antiphase forcing case appears similar to that of the single cylinder. However, in the in-phase condition, although there is a similarity in the trend at moderate to high frequencies, there is a completely opposite behaviour at resonant frequencies. At $FR = 0.8$, the value of drag shoots up to more than three times the value of the stationary case. The wake structure corresponding to this condition is the 2P in-phase mode (see part I in figure 26). A larger wake width with an increased deficit plus higher convection velocity of the vortices (observed during the flow visualisation) present in this case leads to such high values of drag. At higher forcing frequencies, there seems to be no appreciable deviation from the stationary case for both in-phase and antiphase cases.

Figure 26. Different wake modes: F, $\tfrac {1}{2}$(P+2S); G, $\tfrac {1}{2}$(2S), single bluff body wake, antiphase; H, 2P double row; I, 2P in-phase; J, P+S; E, far-wake single vortex street.

At a much higher value of $T/D = 4.0$, the combined effect of the two cylinders is such that a minimum drag condition is achieved, that is, for $FR \ge 2.5$. It is also noted that the phase of oscillation does not seem to have any pronounced effect. This suggests that the coupled vortex wake structure which is prevalent at $T/D = 4.0$ is most susceptible to the effects of rotational oscillation when compared with other spacing values as far as drag reduction is concerned. Even at resonant frequencies the magnitude of drag increment is relatively lesser at this value of spacing. At higher forcing frequencies, the two vortex streets in the far wake interact in a manner that leads to an effectively smaller wake deficit in addition to containing negligible amounts of velocity fluctuation which results in low value of drag.

In most of the cases it can be seen that the drag coefficient as per (3.6) and (3.7) agrees well with the direct force measurement values with some exceptions occurring may be due to uncertainties in the experiments and the assumption of a two-dimensional wake in the control volume analysis.