4.1 Tandem and side-by-side arrangements

There have been quite a few reports on $St$ in the wake of two tandem $({\it\alpha}=0^{\circ })$ or side-by-side $({\it\alpha}=90^{\circ })$ cylinders. The present data, including $St_{0}$ measured in an isolated cylinder wake $(Re=1.3\times 10^{4})$ , are compared with those previously reported in figure 3. The present $St_{0}\,(=0.134)$ agrees well with those reported in the literature (e.g. Vickery Reference Vickery1966; Norberg Reference Norberg1993; Zhou & Antonia Reference Zhou and Antonia1994). So do those for ${\it\alpha}=0^{\circ }$ and $90^{\circ }$ (figure 3).

Figure 3.

Variations in $St$ with $P^{\ast }$ : (a) tandem $({\it\alpha}=0^{\circ })$ and (b) side-by-side $({\it\alpha}=90^{\circ })$ arrangements. Present data were measured at $(x^{\ast },y^{\ast })=(4,-4)$ and (4, 4).

At ${\it\alpha}=0^{\circ }$ , the $St$ drops slowly until $P^{\ast }=3.5$ and then rises gradually, approaching $St_{0}$ , as $P^{\ast }$ increases (figure 3 a). It may be inferred that the critical spacing $P_{cri}^{\ast }$ for transition from reattachment to the co-shedding regime is presently between 3.5 and 4.0 (Liu & Chen Reference Liu and Chen2002), slightly deviating from Sakamoto et al.’s (Reference Sakamoto, Haniu and Obata1987) estimate $(P_{cri}^{\ast }=4.0)$ . The deviation is ascribed to different experimental conditions. The present $Re$ , blockage ratio and $T_{u}$ are $1.3\times 10^{4}$ , 6.3 % and 0.5 %, respectively, compared with $5.52\times 10^{4}$ , 9.8 % and 0.19 % in Sakamoto et al.’s (Reference Sakamoto, Haniu and Obata1987) measurements. In general, a larger blockage ratio and a smaller $T_{u}$ result in a longer vortex formation length (Lee Reference Lee1975; West & Apelt Reference West and Apelt1982; Zdravkovich Reference Zdravkovich1997) and hence a larger $P_{cri}^{\ast }$ , as seen from Sakamoto et al.’s (Reference Sakamoto, Haniu and Obata1987) data. Sakamoto & Haniu (Reference Sakamoto and Haniu1988a ) conducted an experiment on the effect of $T_{u}$ on the wake of two tandem square cylinders and observed that an increased $T_{u}$ led to a smaller $P_{cri}^{\ast }$ , for example, $P_{cri}^{\ast }=4.0$ for $T_{u}<0.19\,\%$ but $P_{cri}^{\ast }=3.2{-}4.0$ for $T_{u}=1.4\,\%$ .

At ${\it\alpha}=90^{\circ }$ (figure 3 b), one single and two distinct $St$ are captured at $P^{\ast }<1.3$ and $1.3<P^{\ast }<2.2$ , respectively. The former occurs when the two cylinders of small $P^{\ast }$ act as a single body and the shear layers separate alternately from the free-stream sides of the cylinders, producing a single Kármán vortex street. On the other hand, the latter is associated with a narrow and a wide street behind the cylinders (Zdravkovich Reference Zdravkovich1987; Alam, Moriya & Sakamoto Reference Alam, Moriya and Sakamoto2003a ; Alam et al. Reference Alam, Sakamoto and Zhou2005; Alam & Zhou Reference Alam and Zhou2013). The higher $St$ is almost three times the lower. At $P^{\ast }=2.2{-}3.0$ , three distinct $St$ are detected, Alam et al. (Reference Alam, Zhou and Wang2011) referred to this $P^{\ast }$ range as a transition regime, where the two- $St$ mode changes intermittently to coupled vortex shedding. While the low and high $St$ are associated with the wide and narrow streets, respectively, the intermediate $St$ corresponds to the coupled streets and is almost equal to $St_{0}$ . At $P^{\ast }>3.0$ , coupled vortex shedding takes place, with $St\approx St_{0}$ . The agreement between the present and published data for ${\it\alpha}=0^{\circ }$ and 90 $^{\circ }$ provides us with confidence in the $St$ data obtained in the wake of two staggered cylinders.

4.2 Staggered arrangement

The $St$ is highly dependent on both $P^{\ast }$ and ${\it\alpha}$ , as is the flow structure (figure 2). Figure 4 presents the dependence of $St$ , measured in both upstream- and downstream-cylinder wakes, on ${\it\alpha}$ for $P^{\ast }=1.5{-}5.0$ , where $St_{0}$ is also given as a reference. We may roughly divide this dependence into three different groups for discussion. At $P^{\ast }=1.5$ , $St$ exhibits a sudden jump in both narrow and wide streets (figure 4 a), apparently distinct from others. The separated shear layer may or may not reattach on the downstream cylinder over $P^{\ast }=2.0{-}3.5$ , producing a sudden jump in $St$ in the upstream-cylinder wake (figure 4 b–e). This reattachment does not occur for $P^{\ast }>3.5$ (figure 4 e–h).

Figure 4.

Variation in $St$ with ${\it\alpha}$ : (a) $P^{\ast }=1.5$ , (b) 2.0, (c) 2.5, (d) 3.0, (e) 3.5, (f) 4.0, (g) 4.5 and (h) 5.0 (○, upstream cylinder; ▿, downstream cylinder; $\cdots \cdots$  , single cylinder). Measurements were made at $(x^{\ast },y^{\ast })=(4,-4)$ and (4, 4). $Re=1.3\times 10^{4}$ .

At $P^{\ast }=1.5$ , a single $St$ , identical in both upstream- and downstream-cylinder wakes, occurs at ${\it\alpha}\leqslant 45^{\circ }$ . The two cylinders behave like a single structure, though the effective width of the two cylinders plays a role, causing a smaller $St$ than $St_{0}$ . At ${\it\alpha}<15^{\circ }$ , $St$ decreases slowly with an increase in ${\it\alpha}$ . Accordingly, the lower shear layer from the upstream cylinder rolls up behind the downstream cylinder without reattachment and the upper shear layer reattaches on the upper surface of the downstream cylinder (figure 2 a), its reattachment point shifting toward the leading corner for higher ${\it\alpha}$ . For ${\it\alpha}=15^{\circ }{-}20^{\circ }$ , the upper shear layer reattaches on the surfaces around the upper leading corner (figure 2 b), resulting in an enhanced mixing in the shear layer and hence an appreciable recovery in $St$ (figure 4 a). As ${\it\alpha}$ increases further from $20^{\circ }$ to $40^{\circ }$ , the shear layer impinges on the front surface of the downstream cylinder (figure 2 c), its impingement position moving away from the upper leading corner to the nominal stagnation point from ${\it\alpha}=20^{\circ }$ to $35^{\circ }$ and then to the lower leading corner from ${\it\alpha}=35^{\circ }$ to $40^{\circ }$ . Accordingly, the flow is retarded and modified to a great degree by the downstream cylinder, taking a longer time to complete one vortex shedding cycle and hence producing a rather rapid drop in $St$ from ${\it\alpha}=20^{\circ }$ to $35^{\circ }$ and a less rapid drop from ${\it\alpha}=35^{\circ }$ to $40^{\circ }$ . Note that a larger ${\it\alpha}$ corresponds to a larger effective body width, which also contributes to the decreasing $St$ from ${\it\alpha}=20^{\circ }$ to $40^{\circ }$ . At ${\it\alpha}=45^{\circ }$ , the shear layer fails to impinge upon the downstream cylinder and flows through the gap (figure 2 d). Once $45^{\circ }<{\it\alpha}\leqslant 65^{\circ }$ , the gap flow gains adequate strength to divide the wake into narrow and wide streets behind the upstream and downstream cylinders, respectively (figure 2 e), associated with the higher and lower $St$ , respectively (figure 4 a). Since a larger ${\it\alpha}$ allows the gap flow to be less biased or allows a growth in the narrow street width, the higher $St$ drops almost linearly, reaching $St_{0}=0.134$ at ${\it\alpha}=65^{\circ }$ . Interestingly, the high and low $St$ correspond to the upstream and downstream cylinders, respectively, for ${\it\alpha}=45^{\circ }{-}65^{\circ }$ but swap their correspondence to the cylinders for ${\it\alpha}>65^{\circ }$ (figure 4 a). The power spectral density functions $E_{u}$ of $u$ measured on the free-stream sides of the two cylinders, i.e. at $(x^{\ast },y^{\ast })=(4,-4)$ and (4, 4), respectively (figure 5 a), indeed show that the higher $St$ , i.e. $f_{2}^{\ast }$ , dominates in the street behind the upstream cylinder at ${\it\alpha}=55^{\circ }$ . The peak at the lower $St$ , i.e. $f_{1}^{\ast }$ , is comparable in strength on both sides, as observed by Hu & Zhou (Reference Hu and Zhou2008). The higher $St$ is almost three times the lower at ${\it\alpha}=90^{\circ }$ . The gap flow may be now biased to either side, that is, the vortex street behind either cylinder can be associated with the lower or higher $St$ , as presented in figure 4(a).

Figure 5.

Typical power spectra, $E_{u}$ , of the streamwise fluctuating velocity $u$ measured at $(x^{\ast },y^{\ast })=(4,-4)$ and (4, 4) for (a) $P^{\ast }=1.5$ , ${\it\alpha}=55^{\circ }$ , (b) 3.0, $5^{\circ }$ , (c) 3.0, $10^{\circ }$ , (d) 4.0, $25^{\circ }$ , (e) 4.0, $35^{\circ }$ , $Re=1.3\times 10^{4}$ . The dashed and solid curves denote upstream and downstream cylinders, respectively.

Consider the case of $P^{\ast }=2.0{-}3.5$ . Given ${\it\alpha}<35^{\circ }{-}40^{\circ }$ , the two cylinders behave like a single bluff body, generating a single Kármán street and hence a single $St$ (figure 4 b–e). The $St$ rises slightly up to ${\it\alpha}=5^{\circ }$ and then decreases. As at $P^{\ast }=1.5$ and ${\it\alpha}=15^{\circ }{-}45^{\circ }$ (figure 4 a), the rise is due to the movement of shear layer reattachment position, which reaches the leading corner of the downstream cylinder at ${\it\alpha}=5^{\circ }$ (e.g. figure 2 g); the decrease in $St$ is linked to the impingement position shifting from the upper to the lower corner, along with an increased effective bluff width for larger ${\it\alpha}$ (Alam & Meyer Reference Alam and Meyer2013). The shear layer goes through the gap, without reattachment, at ${\it\alpha}>30^{\circ }$ , $25^{\circ }$ and $20^{\circ }$ for $P^{\ast }=2.0$ , 2.5 and 3.0, respectively. The gap flow, deflected toward the upstream cylinder, is weak and does not play a significant role in the dynamics over ${\it\alpha}=30^{\circ }{-}35^{\circ }$ at $P^{\ast }=2.0$ , $25^{\circ }{-}40^{\circ }$ at $P^{\ast }=2.5$ and $20^{\circ }{-}30^{\circ }$ at $P^{\ast }=3.0$ . For a higher ${\it\alpha}$ , the gap flow acts as a base bleed and prolongs the formation length (e.g. Zhou et al. Reference Zhou, Wang, So, Xu and Jin2001; Alam & Meyer Reference Alam and Meyer2011), thus increasing $St$ (Alam et al. Reference Alam, Moriya and Sakamoto2003a , Reference Alam, Zhou and Wang2011), but the increased effective bluff width reduces $St$ (Alam & Sakamoto Reference Alam and Sakamoto2005). The two effects tend to cancel each other and $St$ varies little. A further increase in ${\it\alpha}$ allows the deflected gap flow to grow in strength, and a narrow and a wide street emerge, corresponding to a higher and lower $St$ , respectively (figure 4 b–d). As ${\it\alpha}$ approaches $90^{\circ }$ , the gap flow appears symmetric about $y^{\ast }=0$ , forming two coupled vortex streets (e.g. figure 2 k), except at $P^{\ast }=2.0$ where the narrow and wide streets persist and the gap flow may be deflected to either side.

Note the occurrence of two $St$ at ${\it\alpha}=10^{\circ }$ and $20^{\circ }$ at $P^{\ast }=3.0$ . The downstream cylinder is subjected to the impingement of the shear layer (figure 2 g), which is unsteady for a relatively large $P^{\ast }$ . The flow physics behind the observation may be understood through examining $E_{u}$ , along with flow visualization data (figure 2). At ${\it\alpha}=5^{\circ }$ the shear layer reattaches very near the upper leading corner of the downstream cylinder (figure 2 g) and $E_{u}$ displays one pronounced peak at $f^{\ast }=0.138$ (figure 5 b). At ${\it\alpha}=10^{\circ }$ (figure 5 c), the pronounced peak occurs at $f_{2}^{\ast }=0.134$ accompanied by another peak at $f_{1}^{\ast }=0.104$ behind both cylinders. Evidently, this shear layer impingement occurs intermittently on the upper leading corner and front surface of the downstream cylinder, and the bi-stable nature of the flow is responsible for $f_{1}^{\ast }$ and $f_{2}^{\ast }$ . However, the shear layer impinges only upon the front surface as ${\it\alpha}$ increases to $15^{\circ }$ (figure 2 h). This is followed by the resumption of a single $St=0.111$ (figure 4 d), which is almost equal to $f_{1}^{\ast }\,(=0.104)$ at ${\it\alpha}=10^{\circ }$ (figure 5 c). This is not coincidental; both frequencies are connected to the shear layer impingement upon the front surface of the downstream cylinder. Interestingly, the second harmonic of $St$ was absent at ${\it\alpha}=5^{\circ }$ but evident behind the upstream cylinder at ${\it\alpha}=15^{\circ }$ . The difference arises from the reattachment/impingement location, i.e. one $({\it\alpha}=5^{\circ })$ on the leading corner of the downstream cylinder and the other $({\it\alpha}=15^{\circ })$ upon the front surface. The latter is associated with the second harmonic peak in $E_{u}$ . Alam et al. (Reference Alam, Moriya, Takai and Sakamoto2003b ) and Alam (Reference Alam2014) observed, in the wake of two tandem circular cylinders at $P^{\ast }=1.4$ , the second harmonic frequency in the power spectrum of the fluctuating pressure on the cylinder surface when the shear layers from the upstream cylinder impinged upon the downstream cylinder, forming a strong quasi-steady vortex between the cylinders. The second harmonic frequency disappeared at $P^{\ast }=3.5$ where the shear layer impingement was greatly weakened. By ${\it\alpha}=20^{\circ }$ , the shear layer either impinges upon the lower leading corner or passes through the gap without reattachment (figure 2 i) and is again bi-stable. $E_{u}$ at ${\it\alpha}=20^{\circ }$ (not shown) was quite similar to its counterpart at ${\it\alpha}=10^{\circ }$ in figure 5(c), showing two significant peaks at $f_{1}^{\ast }$ and $f_{2}^{\ast }$ and another peak at $2f_{2}^{\ast }$ only behind the upstream cylinder. There are differences between the two cases though. Firstly, while reattaching/impinging intermittently on the upper leading corner and front surface of the downstream cylinder at ${\it\alpha}=10^{\circ }$ , the unsteady shear layer intermittently impinges upon the front surface near the lower leading corner and flows through the gap, without reattachment, at ${\it\alpha}=20^{\circ }$ . Secondly, the higher and lower $St$ are associated with the shear layer reattachment on the upper leading corner and front surface, respectively, at ${\it\alpha}=10^{\circ }$ but are connected to reattachment on the lower corner and the non-reattached flow, respectively, at ${\it\alpha}=20^{\circ }$ .

At $P^{\ast }=3.5$ (figure 4 e), the shear layer impinges upon the front surface for ${\it\alpha}\leqslant 5^{\circ }$ and a single $St$ is observed. At ${\it\alpha}=10^{\circ }$ and $15^{\circ }$ , two different $St$ are observed, due to intermittent shear layer reattachment and non-reattachment. At ${\it\alpha}\geqslant 20^{\circ }$ , both inner shear layers go through the gap, splitting the wake into one narrow and one wide street behind the upstream and downstream cylinders, respectively. Gaining a large momentum under large $P^{\ast }$ , the gap flow is less deflected (Wang & Zhou Reference Wang and Zhou2005; Alam & Zhou Reference Alam and Zhou2013), implying a rather limited departure in width between the two streets and hence a small difference between the high and low $St$ . For larger ${\it\alpha}$ , beyond $20^{\circ }$ , the lateral and longitudinal separations between the cylinders increase and decrease, respectively. The former tends to reduce the difference between the two $St$ , while the latter acts to enlarge it, narrowing the street behind the upstream cylinder. The two effects tend to cancel out each other, maintaining the difference almost unchanged up to ${\it\alpha}=60^{\circ }$ . At ${\it\alpha}>60^{\circ }$ , the latter effect retreats gradually, leading to a collapse of the two $St$ to $St_{0}$ at ${\it\alpha}\geqslant 80^{\circ }$ .

For $P^{\ast }\geqslant 4.0$ , the shear layers roll up, forming vortices, before reaching the downstream cylinder. At ${\it\alpha}=0^{\circ }$ , vortex shedding from the downstream cylinder is triggered by the arrival of vortices from the upstream cylinder, implying the same $St$ in the gap as that behind the cylinders. At ${\it\alpha}>0^{\circ }$ , the upper-row vortices from the upstream cylinder impinge on the front (e.g. figure 2 l) or lower surface (e.g. figure 2 m) of the downstream cylinder, depending on ${\it\alpha}$ . Their direct interaction with shear layers around the downstream cylinder results in synchronized shedding from the two cylinders and a single $St$ , which prevails up to ${\it\alpha}=30^{\circ }$ , $35^{\circ }$ and $40^{\circ }$ for $P^{\ast }=4.0$ , 4.5 and 5.0, respectively (figure 4 f–h). Typical $E_{u}$ is presented in figure 5(d). The ${\it\alpha}$ range of the single $St$ is enlarged at larger $P^{\ast }$ because the two rows of vortices from the upstream cylinder grow laterally downstream. The $St$ is weakly dependent on $P^{\ast }$ and is nearly constant up to at least ${\it\alpha}=15^{\circ }$ , linked to the vortex impingement on the front surface, before a slow rise with ${\it\alpha}$ when the vortices hit the lower surface of the cylinder. At ${\it\alpha}=30^{\circ }{-}75^{\circ }$ , $35^{\circ }{-}70^{\circ }$ and $0^{\circ }{-}70^{\circ }$ for $P^{\ast }=4.0$ , 4.5 and 5.0, respectively, the upstream-cylinder-generated vortex street cannot trigger vortex shedding from the downstream cylinder anymore, and is narrowed under the squeezing effect of the downstream cylinder, resulting in a higher $St$ , as illustrated in figure 5(e). The difference in $St$ between the two streets contracts for higher $P^{\ast }$ where the street behind the upstream cylinder is less squeezed. A further increase in ${\it\alpha}$ leads to negligibly small interference between the streets, their $St$ values collapsing to $St_{0}$ .

4.3 Dependence of $St$ on $P^{\ast }$ and ${\it\alpha}$

The overall picture of $St$ may be obtained via examining the dependence of $St$ on $P^{\ast }$ and ${\it\alpha}$ . Under very complicated and complex interactions between the shear layers, between shear layers and vortices and between vortices, the flow structure may evolve downstream very rapidly, particularly in the near wake. Naturally, the measured $St$ may vary with $x^{\ast }$ as well as from one street to another. It has been found that $St$ obtained at $x^{\ast }=4$ can be different from that at $x^{\ast }=7$ , 10 and 15, though $St$ remains unchanged for $x^{\ast }\geqslant 7$ . As such, the data measured at $x^{\ast }=4$ and 10 are presented and discussed below.

At $x^{\ast }=4$ , $St$ behind the upstream cylinder is 0.10–0.13 at ${\it\alpha}<20^{\circ }$ (figure 6 a), less than $St_{0}\,(=0.134)$ . This is because the downstream cylinder obstructs the wake of the upstream cylinder when $P^{\ast }\geqslant 4$ or directly interacts with the shear layer around the upstream cylinder when $P^{\ast }<4$ , widening the combined wake. $St$ is thus identical to that behind the downstream cylinder (figure 6 b). At $P^{\ast }<4$ , the outer shear layers of the two cylinders roll up, forming vortices at the same $St$ , and the cylinders behave like a single bluff body. On the other hand, at $P^{\ast }\geqslant 4$ a single $St$ is achieved due to vortex shedding from the downstream cylinder triggered by the arrival of the upstream-cylinder-generated vortices.

Figure 6.

Isocontours of $St$ : (a) $x^{\ast }=4$ , upstream cylinder; (b) 4, downstream; (c) 10, upstream; (d) 10, downstream. The $\text{contour increment}=0.01$ . $Re=1.3\times 10^{4}$ . (e) Dependence of the flow regime on $P^{\ast }$ and ${\it\alpha}\,(Re=1.3\times 10^{4})$ . Alam et al.’s (Reference Alam, Zhou and Wang2011,  $Re=4.7\times 10^{4}$ ) data ( $A$ , $B$ , $C$ , $D_{1}$ , $D_{2}$ ) for the side-by-side arrangement and Sakamoto et al.’s (Reference Sakamoto, Haniu and Obata1987,  $Re=2.76\times 10^{4}$ ) data (reattachment and co-shedding regimes) for the tandem cylinders are also included. $A$ : single-body regime, $B$ : two-frequency regime, $C$ : transition regime, $D_{1}$ : antiphase-dominated two-street regime, $D_{2}$ : antiphase-inphase-mixed two-street regime.

For $1.5<P^{\ast }<3.0$ , $St$ drops to 0.07–0.08 and is identical from one street to another given ${\it\alpha}=25^{\circ }{-}40^{\circ }$ (figure 6 a,b). With a weak gap flow, vortex shedding from the free-stream sides of the two cylinders dominates the wake, leading to a single $St$ , which varies little with ${\it\alpha}$ and $P^{\ast }$ . With a further increase in ${\it\alpha}$ or $P^{\ast }$ , the formation of a narrow and a wide street leads to a higher $St$ behind the upstream cylinder (figure 6 a) than behind the downstream (figure 6 b). The former drops rapidly for larger ${\it\alpha}$ or $P^{\ast }$ , while the latter goes up, both reaching $St_{0}$ for ${\it\alpha}>70^{\circ }{-}85^{\circ }$ or $P^{\ast }>2.5$ . The highest $St\,({\approx}0.25)$ or $St/St_{0}\,(=1.87)$ associated with the upstream cylinder occurs at $P^{\ast }=2.0$ and ${\it\alpha}=55^{\circ }$ . So does the lowest $St\,({\approx}\,0.07)$ or $St/St_{0}\,(=0.54)$ in the wake of the downstream cylinder. This means that the downstream cylinder at this configuration can squeeze the upstream-cylinder wake most effectively, producing the largest difference in $St$ between the two streets. In the wake of two circular cylinders, Kiya et al. (Reference Kiya, Arie, Tamura and Mori1980,  $Re=1.58\times 10^{4}$ ) observed the highest $St=0.39$ or $St/St_{0}=1.95$ and lowest $St=0.11$ or $St/St_{0}=0.55$ behind the upstream and downstream cylinders, respectively, at ${\it\alpha}=35^{\circ }{-}45^{\circ }$ and $P^{\ast }\approx 1.8$ . The highest or lowest $St/St_{0}$ and corresponding $P^{\ast }$ are quite close for the two cases, though ${\it\alpha}$ is larger in the case of square cylinders. This is not unexpected in view of the fact that both flow separation angle and wake width are larger in the wake of square cylinders than in the wake of circular cylinders. A further increase in ${\it\alpha}$ or $P^{\ast }$ leads to a diminishing difference in $St$ between the streets until $St=St_{0}$ .

At $P^{\ast }\leqslant 1.5$ and ${\it\alpha}>70^{\circ }$ , the higher and lower $St$ correspond now to the downstream and upstream cylinders, respectively. As ${\it\alpha}$ increases, the higher $St$ drops, while the other remains almost unchanged.

At $x^{\ast }=10$ (figure 6 c,d), the dependence of $St$ on ${\it\alpha}$ and $P^{\ast }$ is overall similar to that at $x^{\ast }=4$ (figure 6 a,b). However, there are two differences. Firstly, the higher $St$ behind the upstream cylinder at $35^{\circ }<{\it\alpha}<65^{\circ }$ for $1.5<P^{\ast }<3.0$ and $20^{\circ }<{\it\alpha}<35^{\circ }$ for $3<P^{\ast }<4$ (figure 6 a) decreases at $x^{\ast }=10$ (figure 6 c). Secondly, the higher $St$ that occurs at ${\it\alpha}>70^{\circ }$ and $P^{\ast }\leqslant 1.5$ behind the downstream cylinder at $x^{\ast }=4$ (figure 6 b) transmutes to the lower at $x^{\ast }=10$ (figure 6 d), $St$ becoming identical in both streets. The observation suggests that the vortices of the higher $St$ in the narrow street, observed at $x^{\ast }\leqslant 4$ , vanish rapidly and are broken up or amalgamated with the vortices of the lower $St$ .

5.1 Mode S-I

Mode S-I is characterized by a single staggered vortex street with a single $St\,(=0.07{-}0.13)$ and is found at ${\it\alpha}<47.5^{\circ }{-}17.5^{\circ }$ depending on $P^{\ast }$ (figure 6). However, there are two types of vortex formation processes, resulting in very different vortex strengths. Thus, this mode is further divided into S-Ia and S-Ib.

Given $P^{\ast }<3.75$ and ${\it\alpha}=0^{\circ }{-}47.5^{\circ }$ , the two cylinders behave like one single body. The inner shear layer separated from the upstream cylinder reattaches on the downstream cylinder at smaller ${\it\alpha}$ or $P^{\ast }$ but goes through the gap at larger ${\it\alpha}$ or $P^{\ast }$ . The outer shear layers separate alternately from the free-stream sides of the cylinders, forming one staggered vortex street or Mode S-Ia. Figure 7(a) presents typical flow structures of PIV-captured instantaneous vorticity contours for mode S-Ia. The maximum vorticity is slightly lower than in a single square cylinder wake (e.g. Lyn et al. Reference Lyn, Einav, Rodi and Park1995; Hu et al. Reference Hu, Zhou and Dalton2006), but the vortex wavelength is longer because of the lower $St$ (figure 6). Note that both positive and negative vortices grow in strength downstream, at least up to $x^{\ast }=15$ . This is a quite typical snapshot. The observation is connected to the way the vortices are generated. Zhou & Yiu (Reference Zhou and Yiu2006) also noted in the wake of two tandem circular cylinders $(L^{\ast }=2.5)$ that the shear layer separated from the upstream cylinder reattached on the downstream side of the downstream cylinder, interrupting the development of the boundary layer and impairing the initial vortex strength. As a result, vortices grew in strength under the effect of mean shear all the way until at least $x^{\ast }=10$ .

Figure 7.

Typical instantaneous PIV-measured vorticity ${\it\omega}_{z}^{\ast }$ contours: (a) S-Ia, $P^{\ast }=2.5$ , ${\it\alpha}=15^{\circ }$ and (b) S-Ib, 4.5, $5^{\circ }$ . $Re=1.3\times 10^{4}$ . The contour cutoff levels are $\pm 0.3$ and the increment is 0.2.

The $E_{u}$ for mode S-Ia displays a pronounced peak at $f^{\ast }=0.111$ (figure 8 a) at $y^{\ast }=0$ , $\pm 1$ , $\pm 3$ and $\pm 5$ . The predominant frequency does not change over $x^{\ast }=4{-}10$ . Furthermore, the characteristics of the power spectrum at $y^{\ast }>0$ are similar to those at $y^{\ast }<0$ , implying a single wake with two rows of vortices at $y^{\ast }>0$ and $y^{\ast }<0$ , respectively, consistent with PIV data (figure 7 a). Peaks at the second and even third harmonics of $St$ are discernible in $E_{u}$ up to $x^{\ast }=10$ (figure 8 c), especially near the wake centreline.

Figure 8.

Typical $E_{u}$ of $u$ measured at (a,b) $x^{\ast }=4$ and (c,d) $x^{\ast }=10$ for mode S-Ia (a,c,  $P^{\ast }=2.5$ , ${\it\alpha}=15^{\circ }$ ) and mode S-Ib (b,d, $P^{\ast }=4.5$ , ${\it\alpha}=5^{\circ }$ ).  $Re=1.3\times 10^{4}$ .

Mode S-Ib occurs at $P^{\ast }>3.75$ and ${\it\alpha}<17.5^{\circ }$ (figure 6 e), featuring vortex shedding from both cylinders at the same $St$ (figure 6). At ${\it\alpha}=0^{\circ }$ , mode S-Ib corresponds to the co-shedding flow (e.g. Sakamoto & Haniu Reference Sakamoto and Haniu1988b ; Ljungkrona, Norberg & Sunden Reference Ljungkrona, Norberg and Sunden1991; Alam et al. Reference Alam, Moriya, Takai and Sakamoto2002; Zhou & Yiu Reference Zhou and Yiu2006), and its ${\it\omega}_{z}^{\ast }$ -contours at $P^{\ast }=4.5$ and ${\it\alpha}=5^{\circ }$ are presented in figure 7(b). Similarly to mode S-Ia (figure 7 a), a staggered vortex street about the centreline is evident in mode S-Ib. However, the initial conditions of modes S-Ia and S-Ib are different. Mode S-Ia is characterized by a smaller $P^{\ast }$ . With the inner shear layer from the upstream cylinder reattaching on the downstream cylinder, the staggered vortices originate largely from alternate shedding from the free-stream sides of the cylinders. On the other hand, S-Ib is characterized by vortex impingement on the downstream cylinder and the staggered vortices behind the cylinders result from interactions between the upstream-cylinder-generated vortices and the upper or lower shear layer, depending on $P^{\ast }$ and ${\it\alpha}$ (figure 6 a–d), around the downstream cylinder. This impingement reduces substantially the vortex strength. For instance, the vortices in figure 7(b) are significantly smaller in both maximum vorticity and size than those in figure 7(a). Hu & Zhou (Reference Hu and Zhou2008) made a similar finding in the wake of two staggered circular cylinders. Furthermore, the upper-row vortices appear weaker than the lower (figure 7 b), as a result of interactions between impinging vortices and the upper shear layer. The PIV results are supported by $E_{u}$ (figure 8 b,d) measured at the same $P^{\ast }$ and ${\it\alpha}$ , which show a pronounced peak at $f^{\ast }=0.119$ in $E_{u}$ , indicating the occurrence of predominant vortices. However, this peak is considerably weaker than that in figure 8(a,c). This peak also decays more rapidly with increasing $x^{\ast }$ for S-Ib than for S-Ia. On another note, this peak is quite evident near the wake centreline for S-Ia but not so for S-Ib. This is because S-Ib consists of weaker and smaller vortices and a larger lateral separation between the two rows of vortices, which is obvious by comparing figure 7(a) with (b).

Note that S-Ia and S-Ib are separated by $P^{\ast }=3.75$ at ${\it\alpha}=0^{\circ }$ (figure 6 e), which is consistent with previous classifications for the wake of two tandem cylinders. Based on the measured force and $St$ at $Re=2.76{-}5.52\times 10^{4}\,({\it\alpha}=0^{\circ })$ , Sakamoto et al. (Reference Sakamoto, Haniu and Obata1987) and Sakamoto & Haniu (Reference Sakamoto and Haniu1988a ) reported $P_{cri}^{\ast }=4.0$ . The $P_{cri}^{\ast }$ was found to be 3.5–3.75 by Luo & Teng (Reference Luo and Teng1990) based on force measurement at $Re=5.67\times 10^{4}$ . It is worthwhile pointing out that the $P_{cri}^{\ast }$ is highly dependent on the free-stream turbulent intensity (Sakamoto & Haniu Reference Sakamoto and Haniu1988a ) and $Re$ as well as how $P^{\ast }$ is varied, i.e. progressively increasing or decreasing (Liu & Chen Reference Liu and Chen2002).

5.2 Mode S-II

Mode S-II occurs in general at large ${\it\alpha}\,({>}17.5^{\circ })$ and small $P^{\ast }\,({<}3.75)$ ; the larger the ${\it\alpha}$ , the smaller the $P^{\ast }$ as long as the lateral separation between the cylinders is adequate for the formation of two counter-rotating vortices in the gap (figure 2 e). Now each cylinder generates one Kármán vortex street, one narrow with a higher $St$ and the other wide with a lower $St$ . Vigorous interactions take place between the two streets, resulting in one single vortex street of the low $St$ at $x^{\ast }>7$ (depending on $Re$ ). The narrow street may occur behind the upstream or downstream cylinder (figure 2 e,f), referred to as S-IIa and S-IIb, respectively.

S-IIa occupies a much larger region than S-IIb (figure 6 e). In this region, the upstream cylinder generates vortices at a higher $St$ at $x^{\ast }<7$ (figure 6 a), but these vortices evolve into a lower $St$ at $x^{\ast }>7$ (figure 6 c). Based on the PIV data (figure 9 a), it is seen that the gap flow is dominated by positive vorticity (induced mainly by the lower side of the downstream cylinder) which can also be predicted from figure 2(e). One may recognize that the merging of two positive-signed vortices, one from the lower side of the upstream cylinder (maximum ${\it\omega}_{z}^{\ast }=1.88$ , lower one) and the other (maximum ${\it\omega}_{z}^{\ast }=1.88$ , upper one) from the gap, enlarges the size of the resultant vortex in the lower street, whose maximum ${\it\omega}_{z}^{\ast }$ drops to 1.48 at $x^{\ast }\approx 12$ . Typical $E_{u}$ of $u$ at $x^{\ast }=4$ and 10 for mode S-IIa are presented in figure 10(a,c) for a representative configuration of $P^{\ast }=2.0$ and ${\it\alpha}=60^{\circ }$ . At $y^{\ast }>0$ , only the peak at $f_{1}^{\ast }=0.067$ and/or its second harmonic is evident at either $x^{\ast }$ , suggesting the occurrence of the predominant vortices in the wide street behind the downstream cylinder. On the other hand, a peak at $f_{2}^{\ast }=0.252$ appears at $y^{\ast }<0$ for $x^{\ast }=4$ (figure 10 a) due to the presence of a downward-deflected narrow street behind the upstream cylinder. Interestingly, a minor peak occurs at $f_{2}^{\ast }$ – $f_{1}^{\ast }$ only for $y^{\ast }<0$ at $x^{\ast }=4$ , reflecting vigorous interactions in the narrow street. The peak at $f_{1}^{\ast }$ is more pronounced at $y^{\ast }>0$ than at $y^{\ast }<0$ , corroborating the observation from figure 9(a).

Figure 9.

Typical instantaneous PIV-measured vorticity ${\it\omega}_{z}^{\ast }$ contours: (a) S-IIa, $P^{\ast }=2.0$ , ${\it\alpha}=60^{\circ }$ and (b) S-IIb, 1.5, $80^{\circ }$ . $Re=1.3\times 10^{4}$ . The contour cutoff levels are $\pm 0.3$ and the increment is 0.2.

Figure 10.

Typical $E_{u}$ of $u$ measured at (a,b) $x^{\ast }=4$ and (c,d) $x^{\ast }=10$ for mode S-IIa (a,c, $P^{\ast }=2.0$ , ${\it\alpha}=60^{\circ }$ ) and mode S-IIb (b,d, $P^{\ast }=1.5$ , ${\it\alpha}=80^{\circ }$ ).  $Re=1.3\times 10^{4}$ .

Mode S-IIb takes place at ${\it\alpha}>67.5^{\circ }$ and $P^{\ast }<1.75$ . In contrast to S-IIa, this mode is characterized by a gap flow deflected toward the downstream cylinder, forming a narrow and a wide street behind the downstream and upstream cylinders, respectively. The flow structure corresponding to mode S-IIb has been presented and discussed earlier (figure 2 f). Figure 9(b) presents the PIV-measured ${\it\omega}_{z}^{\ast }$ -contours at $P^{\ast }=1.5$ and ${\it\alpha}=80^{\circ }$ . The vorticity concentrations associated with the gap flow are discernible, echoing the change in the gap flow direction from the lower to the upper (figure 2 f). The presence of the positive vorticity, with a maximum of 0.51, behind the downstream cylinder bears the signature of the gap flow being biased upward. Typical $E_{u}$ (figure 10 b,d) at $P^{\ast }=1.5$ and ${\it\alpha}=80^{\circ }$ exhibits a pronounced peak at $f_{1}^{\ast }=0.063$ across the wake at $x^{\ast }=4$ and 10 and another broad peak at $f_{2}^{\ast }=0.193$ only at $x^{\ast }=4$ and $y^{\ast }>0$ (figure 10 b). Apparently, the peak at $f_{1}^{\ast }$ is more pronounced at $y^{\ast }<0$ than at $y^{\ast }>0$ . The observations are consistent with the flow visualization data (figure 2 f).

Figure 11.

Typical instantaneous PIV-measured vorticity ${\it\omega}_{z}^{\ast }$ contours at $P^{\ast }=4.0$ , ${\it\alpha}=55^{\circ }$ ; mode T-I. $Re=1.3\times 10^{4}$ .

Figure 12.

Typical $E_{u}$ of $u$ measured at (a) $x^{\ast }=4$ and (b) $x^{\ast }=10$ for mode T-I ( $P^{\ast }=4.0$ , ${\it\alpha}=55^{\circ }$ ).  $Re=1.3\times 10^{4}$ .

5.3 Mode T-I

Mode T-I occurs at a larger $P^{\ast }$ or ${\it\alpha}\,(P^{\ast }>1.75,32.5^{\circ }<{\it\alpha}<90^{\circ })$ than S-II. A large $P^{\ast }$ or ${\it\alpha}$ implies a large momentum of the gap flow and also a large separation between the two counter-rotating gap vortices. Consequently, the narrow and wide streets do not merge downstream and, instead, retain their identities with high and low $St$ , respectively. With a variation in $P^{\ast }$ or ${\it\alpha}$ , the higher $St$ behind the upstream cylinder drops from 0.22 to 0.13, but the lower $St$ behind the downstream cylinder increases from 0.08 to 0.13 (figure 6). As illustrated in figure 11, the two vortex streets appear to be qualitatively the same. Actually, the upstream-cylinder-generated street is narrower than the downstream-cylinder-generated one. A smaller streamwise separation between the two consecutive vortices in the narrow street compared to that in the wide street suggests a higher $St$ . Both streets display staggered vortices, with an appreciably larger size and strength of vortices for the upper street. With a lower frequency, vortex shedding from the downstream cylinder may have more time, than that from the upstream cylinder, to accumulate vorticity (Zdravkovich Reference Zdravkovich1985), yielding the vortices of a larger size and strength and hence a wider street. Naturally, mode T-I is asymmetric about $y^{\ast }=0$ . The interactions between the two streets of different $St$ however accelerate the downstream decay of vortices (Sumner et al. Reference Sumner, Price and Paidoussis2000; Zhou, Zhang & Yiu Reference Zhou, Zhang and Yiu2002; Alam & Zhou Reference Alam and Zhou2013). It will be shown later that two antiphased streets of an identical $St$ persist longer downstream. The two distinct predominant frequencies in the two streets are confirmed by $E_{u}$ (figure 12) at $P^{\ast }=4.0$ and ${\it\alpha}=55^{\circ }$ . Two pronounced peaks occur at $f_{1}^{\ast }=0.126$ and $f_{2}^{\ast }=0.152$ above and below the centreline, respectively, in the downstream- and upstream-cylinder-generated streets. The two frequencies are lower and higher than that (0.134) in an isolated square cylinder wake, respectively. Note that both peaks at $f_{1}^{\ast }$ and $f_{2}^{\ast }$ may occur over $y^{\ast }=-1{-}+1$ , that is, both predominant frequencies are detectable in the inner rows of the two streets, in contrast to the observation in S-IIb where the low $St$ occurs across the narrow street as well (figure 10). The peaks at $f_{1}^{\ast }$ and $f_{2}^{\ast }$ decay rather rapidly for larger $x^{\ast }$ , as a result from the interactions between the two streets of distinct frequencies as well as the mechanisms associated with an isolated cylinder wake. Landweber (Reference Landweber1942) analysed the stability of two streets based on a model of infinitely long rows of point vortices, similarly to the classical von Kármán analysis of a single vortex street (Zdravkovich Reference Zdravkovich1997). He came to the conclusion that, given two stable streets, the convection velocity must be the same for all vortices and the two streets must be either antiphased or inphased. As such, it may be inferred that if two streets differ in width, their phase shift will change and the flow structure will be unstable, accelerating the decaying rate.

5.4 Mode T-II

The difference in $St$ between the narrow and wide streets dwindles as $P^{\ast }$ increases (figure 4). For $P^{\ast }>2.75$ and ${\it\alpha}>72.5^{\circ }$ , this difference is no more than 1 %, comparable to the experimental uncertainty. Apparently, such a flow is different from T-I and is called T-II. Mode T-II occurs in two regions of the $P^{\ast }$ – ${\it\alpha}$ plane (figure 6 e), i.e. mode T-IIa at large ${\it\alpha}\,({>}72.5^{\circ })$ and $P^{\ast }>2.75$ and mode T-IIb at $17.5^{\circ }\leqslant {\it\alpha}<42.5^{\circ }$ and large $P^{\ast }\,({>}3.75)$ . In mode T-II, the predominant frequencies in both streets are identical to each other and almost the same as that (0.134) in an isolated cylinder wake. Vortices shed from the two cylinders are synchronized, either antiphased or inphased. T-IIa and T-IIb are distinguished from each other based on their different synchronization mechanisms and initial conditions.

Mode T-IIa resembles the flow structure behind two side-by-side cylinders (Alam et al. Reference Alam, Zhou and Wang2011). The two cylinders generate individual streets of the same shedding frequency and a similar width. Typical instantaneous ${\it\omega}_{z}^{\ast }$ contours at $P^{\ast }=4.5$ and ${\it\alpha}=85^{\circ }$ are illustrated in figure 13(a). The wake consists of two Kármán vortex streets of the same width and frequency, which are either antiphased (figure 13 a) or inphased (not shown). The ${\it\omega}_{z}^{\ast }$ -contours indicate comparable vortex strengths between the two coupled streets, which are essentially the same as in an isolated square cylinder wake (Hu et al. Reference Hu, Zhou and Dalton2006). The observations are further supported by $E_{u}$ (figure 14 a,c) measured at $P^{\ast }=4.5$ and ${\it\alpha}=85^{\circ }$ , which shows a pronounced peak at $St=0.134$ across the wake at both $x^{\ast }=4$ and 10. A comparison in $E_{u}$ between figures 12 and 14 indicates that T-IIa is more stable than T-I. The coupling between the streets is largely dictated by vortex shedding associated with the gap flow (Alam et al. Reference Alam, Moriya and Sakamoto2003a ; Alam & Zhou Reference Alam and Zhou2013) and the antiphased shedding overwhelms the inphased. The inphased vortex shedding occurs occasionally based on the PIV data. Alam et al. (Reference Alam, Zhou and Wang2011) observed in the wake of two side-by-side square cylinders approximately 65 % of antiphased shedding at $P^{\ast }=3.0$ . This percentage dropped to 25 % at $P^{\ast }=5.0$ . On the other hand, the inphased shedding was 2 % and 14 % at $P^{\ast }=3.0$ and 5.0, respectively. Kamemoto (Reference Kamemoto1976, $Re=3\times 10^{4}$ ) and Alam et al. (Reference Alam, Moriya and Sakamoto2003a ,  $Re=5.5\times 10^{4}$ ) showed in the wake of two side-by-side circular cylinders that vortices shed from the two cylinders were predominantly antiphased at $P^{\ast }=2.1{-}3.5$ but both antiphased and inphased at $P^{\ast }>3.5$ . Alam & Zhou (Reference Alam and Zhou2013) showed at $Re=300$ that the antiphased street persists for a longer $x^{\ast }$ than the inphased.

Figure 13.

Typical instantaneous PIV-measured vorticity ${\it\omega}_{z}^{\ast }$ contours: (a) T-IIa, $P^{\ast }=4.5$ , ${\it\alpha}=85^{\circ }$ and (b) T-IIb, 5.0, $30^{\circ }$ . $Re=1.3\times 10^{4}$ . The contour cutoff levels are $\pm 0.3$ and the increment is 0.2.

Figure 14.

Typical $E_{u}$ of $u$ measured at (a,b) $x^{\ast }=4$ and (c,d) $x^{\ast }=10$ for mode T-IIa (a,c, $P^{\ast }=4.5$ , ${\it\alpha}=85^{\circ }$ ) and mode T-IIb (b,d, $P^{\ast }=5.0$ , ${\it\alpha}=30^{\circ }$ ).  $Re=1.3\times 10^{4}$ .

At mode T-IIb, two streets are generated and the gap vortex separated from the upstream cylinder continues to trigger the gap vortex shedding from the downstream cylinder, thus resulting in coupled vortex shedding, at the same $St$ , from both cylinders. This mechanism is different from mode T-IIa, where vortex shedding from one cylinder is either in phase or in antiphase with that from the other cylinder, which is essential for the two streets to be stable. As illustrated by the ${\it\omega}_{z}^{\ast }$ -contours (figure 13 b), the closely approached counter-rotating vortices in the two inner rows are appreciably weaker in strength than those in the outer rows, largely due to the cancelation of positive and negative ${\it\omega}_{z}^{\ast }$ in their interactions (Zhou et al. Reference Zhou, Zhang and Yiu2002). Nevertheless, both streets, largely antiphased, persist far downstream, as the case of T-IIa; the pronounced peak at $St=0.13$ is evident in $E_{u}$ across the wake at both $x^{\ast }=7$ and 10 (figure 14 b,d).

A remark is due on the comparison between the present regimes and those in Alam et al. (Reference Alam, Zhou and Wang2011). Their regime $A\,(P^{\ast }<1.3)$ is a single Kármán vortex street of low frequency where vortices are separated only from the free-stream sides of the cylinders. Regime $B$ , consisting of a narrow and a wide street of distinct $St$ , corresponds to part $({\it\alpha}>80^{\circ })$ of S-IIb and part of T-I. Almost one half of their regime $C$ , where flow modes $B$ and $D$ occur intermittently, corresponds to mode T-I and the other half to T-IIa. Regime $D$ is included in T-IIa.