Flow characteristics and dynamic responses of a rear circular cylinder behind the square cylinder with different side lengths

2.1. Governing equations and numerical details

To account for the fluid dynamics in the VIV and WIV systems, the non-dimensional Navier-Stokes (N-S) equation and the continuity equation for the incompressible viscous flow are employed, shown in Eqs. (1-2). Both equations are described under the ALE framework [24]:

where $u_i$ represents the fluid velocity, and $w_i$ the grid velocity tensor in the $i$th direction, $p$ the pressure, $x_i$ ($i=$ 1, 2 in 2-D problem and 1, 2, 3 in 3-D problem) the Cartesian coordinates, and $t$ the time. Further, the Reynolds number is $Re=$$\left(U_{\infty }D\right)/v$, where $D=$ 1.0 is the characteristic length in the non-dimensional fluid domain, $U_{\infty }$ the free stream velocity, and $v$ the fluid kinematic viscosity.

The computation procedure forms into a semi-implicit CBS scheme, which is expected to be more stable and robust than the classical explicit one [21, 26, 27]. Details of the CBS algorithm have been shown in our previous publication [21]. Some essential equations of the algorithm are listed as below:

The movement governing equations of the oscillating cylinder with two-degrees-of-freedom (2-DOF) are modeled as a mass-spring damper system. The solid dynamic Eqs. (6-7) can be solved by a Newmark-$\beta$ type method introduced in the work by Bao et al. [12]. The discrete forms of the computational steps are also presented in Han et al. [21]:

where, the $X$ and $Y$ are the cylinder displacements in the $x_1$- and $x_2$- direction. $M_r=$$4m/\left(\pi \rho D^2\right)$ is the cylinder’s non-dimensional reduced mass, where $m$ is the cylinder mass of a unit length, and $\rho$ is the fluid density. $\xi$ is the structural damping ratio. $U_r=U_{\infty }/\left(f_{n}D\right)$ represents the reduced velocity, where $f_n$is the natural frequency of the cylinder. $C_L=\left(2F_L\right)/\left(\rho U_{\infty }^{2}D\right)$ and $C_D=\left(2F_D\right)/\left(\rho U_{\infty }^{2}D\right)$ are the lift and drag coefficients, as $F_L$ and $F_D$ are lift and drag force applied on the cylinder, respectively.

The cylinder motion within a time step will induce the updating of the grid. The grid displacement, $W_i$, is calculated using a Laplace-related algorithm shown as follows:

where $\gamma$ is a parameter for controlling the mesh deformation. The grid velocity is calculated by $w_i=(\partial W_i)/\partial t$. Eq. (8) can be solved by standard FEM procedure. Details of the grid updating scheme is introduced in Wang et al. [20].

For solving the VIV case which is one of the typical fluid-structure interaction (FSI) problems, the staggered algorithm is employed. This scheme has been used for satisfactory solutions as shown in the publications [12, 20, 28]. Briefly, the computational steps during one time interval $∆t=t^{n+1}-t^n$ are summarized as: (a) obtain the velocities and pressure in the fluid domain by the CBS algorithm; (b) calculate the drag and lift coefficients, $C_D^{n+1}$ and $C_L^{n+1}$, of the cylinder, and compute the cylinder motions, $X^{n+1}$ and $Y^{n+1}$, by the Newmark-$\beta$ type method; (c) get the updated grid movements, $w_i^{n+1}$, using the equation; (d) return to step (a) to restart a new iteration step until the VIV behaviors become stable.

2.2. Code verification

The computational code has been verified in our previous work [21, 29]. We further validate the accuracy of the computational approach by using it to solve the VIV of an isolated cylinder with two degrees of freedom in uniform cross-flow. This example has also been taken as a reference by Papaioannou et al. [17]. The free oscillation of a circular cylinder with 2-DOF is carried out in the subsection. The size of flow domain, the geometry of the model and boundary conditions are given in Fig. 1(a). The cylinder whose center locates at point (0, 0) has a diameter $D=$ 1.0. The entire computational domain is defined as $\Omega =$[–25$D$, 54$D$]×[–30$D$, 30$D$]. On the surface of the cylinder it is imposed as $u=\dot{X}$, $v=\dot{Y}$ and a uniform flow used at the inlet of the computational domain as a boundary condition is prescribed to be $u_x=U_{\infty }=$ 1.0 and $u_y=$ 0.0, while the pressure is set to zero at the outlet. Symmetry conditions are used at the lateral boundaries as: $\partial u_x/\partial y=$ 0.0, $u_y=$ 0.0. The system parameters are set as follows: the reduced mass $M_r=$ 10.0, the damping ratio $\xi =$ 0.01, the reduced velocity 3.5$\le U_r\le$10.0, the damping ratio Reynolds number $Re=$ 160 and $∆t=$ 0.002.

The schematic mesh grids for the whole computational domain and around the cylinder are respectively illustrated in Figs. 1(b) and 1(c). In the wake flow region and the area around the cylinder, higher resolutions of grid density are acquired. To test the mesh independency, a mesh-refinement study is performed. Three groups of grid system are adopted in the current examples. In Grid I, there are 100 nodes on the surface of the cylinder, while in Grid II and Grid III there are 200 and 400 nodes, respectively. Some details of the three groups are presented in Table 1, which also provide the results of some grid parameters, with the percentage changes written inside the bracket. It is observed that the maximum deviation of results, such as $X_{rms}$ and $Y_{max}$, is 6.03 %, when comparing Grid I with Grid II, whereas no significant change exists between Grid II and Grid III (only 2.52 % change in the value of $X_{max}$). Therefore, all the investigations presented hereafter are based on the mesh characteristics of Grid II.

Fig. 1Schematic of an isolated oscillating circular cylinder

a) The computational domain and boundary conditions

b) The whole mesh

c) Meshes around the cylinder

Comparisons in the non-dimensional maximum amplitudes ($X_{max}/D,Y_{max}/D$) in 2-DOF are shown in Fig. 2 between our results and existing data [17]. As shown in Fig. 2, these results are in good accordance with each other, verifying that the numerical scheme and resolution used in this paper are adequately accuracy and reliable for the solution of VIV in the laminar region.

Fig. 2Comparisons on the dimensionless maximum vibration amplitudes from present numerical results with existing data (Papaioannou et al., [17]) for a 2-DOF oscillating circular cylinder

a) Inline direction ($X_{max}/D$)

b) Transverse direction ($Y_{max}/D$)

4.1. Flow patterns

Fig. 4 summarizes the wake patterns to show how they are affected by the $d/D$ and $U_r$. Three types of fully developed flow pattern, 2S, 2P and 2T, are observed at different side length ratios ($d/D$) and reduced velocities ($U_r$). At lower $d/D$, the 2S and 2P mode are dominant. The vortex street is not formed in the gap between both cylinders at $d/D=$ 0.2, as seen in Figs. 5(a) and (b). With the increasing of $d/D$, vortex shedding phenomenon is clearly seen behind the square cylinder, as seen in Figs. 5(c)-(e). At $d/D=$ 1.0, 2$P$ and 2$S$ modes are observed at 3 $\le U_r\le$ 7 and 9 $\le U_r\le$ 26, respectively. 2T mode occurs at $d/D=$ 1.0 and $U_r=$ 8, as seen in Figs. 5(d). On the contrary, as $d/D=$ 1.5, 2P mode occurs at higher $U_r$ (9 $\le U_r\le$ 26) and 2S mode shows at lower $U_r$. At $d/D=$ 2.0, the predominant vortex shedding patterns are 2S and 2P mode. It is worth noting that a pair of vortices sheds from both upper and lower sides of the rear cylinder, as seen in Fig. 5(e), and the shape of vorticity behind the rear circular cylinder resembles the figure-eight, then two vortices gradually merge into a big one.

Fig. 3Computational model and boundary conditions for the oscillating circular cylinder behind a stationary square cylinder

Fig. 4Flow pattern distributions at different side length ratios (d/D) and reduced velocities (Ur) for the WIV of the circular cylinder: ∆ = 2S pattern (solid and open squares denote steady and wake in the gap between cylinders, respectively.); ◇= 2S pattern with the figure-eight shape of vorticity; ∆ = 2P pattern (solid and open triangles denote steady and wake in the gap between cylinders, respectively.); and ○ = 2T pattern

4.2. Frequency of oscillation

Fig. 6 shows the relationships between frequency ratios ($f_x/f_n$, $f_y/f_n$) and reduced velocities related to the WIV systems. The definitions of these frequencies are given as follows: $f_n$, the natural frequency of the circular cylinder; $f_x$ and $f_y$: the main vibrating frequency of the circular cylinder in the $x_1$- and $x_2$- direction, respectively.

As shown in Fig. 6(a), when the reduced velocity is small (3 $\le U_r\le$ 9), the variations of frequency ratio as a function of reduced velocity are obvious with the increasing of size ratio in the inline direction. At $d/D=$ 0.2, the $f_x/f_n$ gradually increases as $U_r$ increases. The inline vibrating frequency, $f_x$, is twice of its natural frequency, 2$f_n$, for 5 $<U_r\le$ 9. For the case of $d/D=$ 0.5, the $f_x/f_n$ gradually decreases with the increasing of $U_r$ and reaches the minimum value at $U_r=$ 5, and then it increases. While, the trend of $f_x/f_n$ at $d/D=$ 1.0 with the change of $U_r$ is reverse to that of the $d/D=$ 0.5 case: it reaches a peak as high as 2.0 at $U_r=$ 7, and then followed by a dramatic decrease at $U_r=$ 8. As the size ratio further increases, the value of $f_x/f_n$ continuously increases with the increasing of $U_r$. Additionally, for the large reduced velocity (9 $<U_r\le$ 26), the trend of $f_x/f_n$ with the change of $U_r$ is the same at different size ratio cases: with the increasing of $U_r$, the value of $f_x/f_n$ gradually increases. It should be noted that, for the case of $d/D=$ 1.0, the value of $f_x/f_n$ is smaller than that of the other cases in the range of 9 $<U_r\le$ 19, and the frequency ratio is close to 1.0 for 14 $\le U_r\le$ 19.

Fig. 5Instantaneous vorticity contours for the WIV system at different cases

a)$d/D=$ 0.2, $U_r=$ 4

b)$d/D=$ 0.2, $U_r=$ 7

c)$d/D=$ 0.5, $U_r=$ 8

d)$d/D=$ 1.0, $U_r=$ 8

f)$d/D=$ 1.5, $U_r=$ 10

e)$d/D=$ 2.0, $U_r=$ 8

Fig. 6The relationships between frequency ratio and reduced velocity for the rear circular cylinder at different ratios of d/D

a) In the $x_1$-direction ($f_x/f_n$)

b) In the $x_2$-direction ($f_y/f_n$)

On the other hand, with the change of $d/D$, there is a change trend of vibration frequency only with small variations at small reduced velocities in the transverse direction. It can be found from Fig. 6(b), the change of $d/D$ has a great influence on the resonance region of downstream cylinder. For the small $d/D$, the transverse vibration frequency is close to the natural frequency in the range of 5 $\le U_r\le$ 9. However, for the large $d/D$, the synchronization phenomenon occurs at 6 $\le U_r\le$ 8 ($d/D=$ 1.0), 9 $\le U_r\le$ 11 ($d/D=$ 1.5), 14 $\le U_r\le$ 16 ($d/D=$ 2.0), respectively.

4.3. Amplitude of oscillation

To understand the effects of the shape and reduced velocity changes on the dynamic responses of the downstream circular cylinder, the relationships of non-dimensional maximum vibrating amplitudes for the circular cylinder under various reduced velocities in different cases are presented in Fig. 7. Additionally, the results of the single cylinder case with similar structural parameters are also plotted for comparison. The $X_{amp}$ is the maximum vibrating amplitude defined as ($X_{max}-X_{min})/\text{2}$ within one vibrating cycle, while $Y_{amp}$ is that of ($Y_{max}-Y_{min})/\text{2}$.

Fig. 7The relationships between the maximum vibrating amplitude and reduced velocity for the circular cylinder at different ratios of d/D

a) In the $x_1$-direction ($X_{amp}/D$)

b) In the $x_2$-direction ($Y_{amp}/D$)

It can be observed from Fig. 7 that the maximum amplitudes of the rear cylinder are very close to those expected for an isolated cylinder at almost the same reduced velocity for the lower $d/D$$(d/D=\text{0.2})$. There is no vortex street behind the square cylinder, as seen in Figs. 5(a) and 5(b). Hence, the wake of the square cylinder has a significant effect on the rear one. The same behavior was also observed in the published literatures [14, 17]. With the increasing of $d/D$, the changes of oscillation amplitude at both the inline and transverse direction are obvious. In the inline direction, the vibration amplitude of circular cylinder is very small at lower reduced velocities. The value of $X_{amp}$ gradually increases until it reaches a peak at $U_r=$ 5 ($d/D=$ 0.5) and $U_r=$ 8 ($d/D=$ 1.0), respectively. Then the values of $X_{amp}$ keep larger in the range of 5 $\le U_r\le$ 8 ($d/D=$ 0.5) and 8 $\le U_r\le$ 12 ($d/D=$ 1.0). The oscillation amplitude in the inline direction gradually decreases and then it remains constant in the higher reduced velocity region. On the other hand, as $d/D=$ 0.5, the maximum amplitude ($Y_{amp}/D$) firstly increases until it reaches the maximum at $U_r=$ 6, and then decreases, and finally maintains a smaller value at the higher reduced velocities ($\text{10}\le U_r\le \text{26}$). The trend of transverse vibration amplitude at $d/D=$ 1.0 is similar to that of $d/D=$ 0.5 case. It should be noted that there is a dramatic increase in the peak value of $Y_{amp}/D$ for every case. It reaches a peak as high as $Y_{amp}/D=$ 0.67 at $U_r=$ 6 and $d/D=$ 0.5, while as $d/D=$ 1.0 the peak value of $Y_{amp}/D$ reaches 1.05 at $U_r=$ 7. For the higher $d/D$, the dynamic characteristics of the circular cylinder only appears the ‘single-resonance’ phenomenon in the transverse direction for $d/D\le$ 1.5, while the performance of ‘dual-resonance’ is found at $d/D=$ 2.0. For $d/D=$ 1.5, the oscillation amplitude in the inline direction is smaller in comparison with the case of $d/D=$ 2.0. However, the trend of $X_{amp}/D$ is similar to that at $d/D=$ 2.0: as $U_r$ continuously increases, the value of $X_{amp}/D$ increases until it reaches a local peak at $U_r=$ 4, then gradually decreases to be a small value, increases again, and finally approaches to a constant. In addition, the peak of the transverse amplitude occurs at $U_r=$ 11 ($d/D=$ 1.5), and $U_r=$ 16 ($d/D=$ 2.0), respectively. It should also be noted that a dramatic change in the $Y_{amp}/D$ can be found at $d/D=$ 2.0. The comparisons between the current case and the other cases additionally demonstrate that parameters, such as spacing ratio, cylinder shape, and the existence of the front cylinder, have a significant influence on the vibrations of the rear circular cylinder.

4.4. $\mathit{X}$-$\mathit{Y}$ trajectories of the rear circular cylinder

In general, the trajectory of an isolated elastically mounted cylinder resembles a figure “8”, while the motion shows the shapes of figure “8”, figure “dual-8” and raindrop for the cylinder behind a stationary square cylinder. The Lissajous figures of the trajectory for two tandem cylinders at different reduced velocities when $d/D=$ 0.2, 0.5, 1.0, 1.5 and 2.0 are presented in Figs. 8-12, respectively.

For the smaller $d/D$($d/D=$ 0.2), the dynamic behaviors are similar to those expected for an isolated cylinder, and the motion orbits of the circular cylinder resemble figure “8” motion, as seen in Fig. 8. It can be seen that, the downstream circular cylinder motions appear to be figure “8” type and its amplitude is quite low in the unlock-in region. However, in the range of lock-in, the oscillation amplitude is large in the transverse direction. It represents the result associated to the ‘single-resonant’ only in the transverse direction.

Fig. 8X-Y trajectories of the 2-DOF circular cylinders behind a square cylinder under different reduced velocities at d/D= 0.2

a)$U_r=$ 3

b)$U_r=$ 5

c)$U_r=$ 8

d)$U_r=$ 9

e)$U_r=$ 10

f)$U_r=$ 12

As $d/D$ increases up to 0.5 and 1.0, the change of $U_r$ has a significant influence on the motion of the downstream cylinder. The Lissajous figures of the trajectory resemble irregular figure at $d/D=$ 0.5, as shown in Fig. 9. The dominant of cylinder orbital trajectories is irregular figure 8 for the small $U_r$, while the motion orbit transforms from the raindrop to a messy and unrepeatable trajectory at $U_r=$ 9. Additionally, at $d/D=$ 1.0, with the increasing of $U_r$, the shape of cylinder motion transforms from the figure “8” to the irregular shape.

For the larger $d/D$ cases ($d/D=$ 1.5 and 2.0), the prominent $X$-$Y$ trajectories of the circular cylinder are figure “8” and figure “dual-8”, as shown in Figs. 11 and 12. As $U_r=$ 4, the motion orbits of downstream cylinder is not symmetrical to the central line. A possible explanation for this feature is that the transverse fluid force on the surface of the cylinder is unbalanced, shown in Fig. 14(c). Additionally, the Lisasajous figure of the circular cylinder exhibits vertical figure “dual-8” at $U_r=$ 8 and $d/D=$ 1.5. Moreover, the motion orbit of the circular cylinder shows the lateral figure “dual-8” shape at $U_r=$ 6 and $d/D=$ 2.0, which is symmetrical to the central line. With the increasing of $U_r$, the oscillation amplitude of the cylinder in the transverse direction gradually increases, then maintains a large value Fig. 7(b), and the shape of cylinder motion transforms from the lateral figure “dual-8” to the figure “8”. As $U_r$ further increases, the motion orbit of the rear cylinder exhibits vertical figure “dual-8” at $U_r=$ 14. And then the shapes of cylinder motion show figure “8” again at larger $U_r$.

Fig. 9X-Y trajectories of the 2-DOF circular cylinders behind a fixed square cylinder under different reduced velocities at d/D= 0.5

a)$U_r=$ 3

b)$U_r=$ 5

c)$U_r=$ 8

d)$U_r=$ 9

e)$U_r=$ 10

f)$U_r=$ 14

Fig. 13 shows the time histories of the motions ($x/D$, $y/D$) and the fluid force coefficients ($C_L$, $C_D$) of the rear circular cylinder, power spectral densities (PSD) of the displacements and the oscillation orbits at different cases. It is worth noting that in this study the motion of the cylinder is sensitive to the size of the upstream square cylinder. With the change of size ratio ($d/D=$ 0.2-2.0), the motion of the cylinder converts from the figure “8” trajectory to the raindrop one for the small $d/D$ case, and then it shows irregular figure “8”; the $X$-$Y$ trajectory resembles figure “dual-8” in the single-resonance response and the response characteristics appear dual-resonance phenomena with a further increasing value of $d/D$.

To explain the effect of the size of the upstream square cylinder on the vibration characteristics of downstream cylinder, the phase portraits of the forces ($C_L$, $C_D$) relative to the vibrations ($x/D$, $y/D$) of the downstream cylinder are presented in Fig. 13. The size ratio has a significant effect on the phase portraits of the forces and the corresponding motions of the cylinder. In the cases of $d/D=$ 0.5 and $U_r=$ 9, the phase portrait of $C_D$ relative to $x/D$ is out-of-phase, giving rise to the smaller-amplitude oscillation in the in-line direction, seen in Fig. 13(a). However, for the case of $d/D=$ 1.0 and $U_r=$ 8, the time of each cycle for the rear cylinder is longer, acting as a modulation period. With the increasing of $d/D$, the phase portraits of fluid forces relative to displacements are similar to that of the case of $d/D=$ 0.5 and $U_r=$ 9, seen in Fig. 13(c). As the $d/D$ further increases to 2.0, the phase portrait of $C_D$ relative to $x/D$ changes from out-of-phase to in-phase, resulting in the larger-amplitude oscillation in the inline direction. Hence, the size ratio of two cylinders causes the changes of the energy transferring process and the oscillation amplitude.

Fig. 10X-Y trajectories of the 2-DOF circular cylinders behind a fixed square cylinder under different reduced velocities at d/D= 1.0

a)$U_r=$ 3

b)$U_r=$ 5

c)$U_r=$ 7

d)$U_r=$ 8

e)$U_r=$ 10

f)$U_r=$ 13

g)$Ur=$ 20

On the other hand, with the change of $d/D$, the PSD features of the vibration displacement singles dramatically change. The above-mentioned VIV responses are explained conveniently in terms of the characteristics of PSD. The multiple vibration frequency components of the displacements at both the inline and transverse direction are co-triggered at the same frequency, resulting in the raindrop motion, as shown in Fig. 13(a). The same phenomenon was also observed in our previous work [28]. With the increasing of $d/D$, the multiple vibration frequency components of the transverse direction with those of the inline one and the dominant vibration frequency in the transverse direction is twice of that in the inline one. Hence, the $X$-$Y$ trajectory shows the irregular figure “8”, as shown in Fig. 13(b).

For the larger size ratio, there are only the 1st and 2nd harmonic contents in the displacement singles at both the inline and transverse direction and the dominant vibration frequency in the transverse direction is twice of that in the inline one, seen in Fig. 13(c). However, the value of PSD relative to the dominant transverse vibration frequency is much greater than that in the inline direction. Hence, the oscillation of the circular cylinder shows signal-resonance of figure “dual-8” motion. When the values of PSD relative to the dominant vibration frequency are the same at both the transverse and inline direction in the case of $d/D=$ 2.0 and $U_r=$ 6, resulting in the inline vibration of the cylinder significantly intensifying and the dual-resonance of figure “dual-8” motion.

Fig. 11X-Y trajectories of the 2-DOF circular cylinders behind a fixed square cylinder under different reduced velocities at d/D= 1.5

a)$U_r=$ 3

b)$U_r=$ 4

c)$U_r=$ 6

d)$U_r=$ 8

e)$U_r=$ 9

f)$U_r=$ 10

g)$Ur=$ 20

4.5. Hydrodynamic coefficients

Due to the long distance between the two cylinders, the effect of the rear cylinder on the front one is insignificant and the upstream square cylinder behaviors are similar to the single cylinder case. Hence, we only select the data of the rear cylinder to display the hydrodynamic characteristics of the WIV system in the subsection. The mean and root mean square (rms) values of the hydrodynamic coefficients, including the drag and lift coefficients ($C_D$ and $C_L$), of the two cylinders are shown in Fig. 14, respectively.

Fig. 12X-Y trajectories of the 2-DOF circular cylinders behind a fixed square cylinder under different reduced velocities at d/D= 2.0

a)$U_r=$ 3

b)$U_r=$ 4

c)$U_r=$ 6

d)$U_r=$ 8

e)$U_r=$ 9

f)$U_r=$ 14

g)$Ur=$ 20

The mean drag coefficient ($C_{D2,mean}$) of the vibrating downstream circular cylinders increases with the reduced velocity until it reaches the peak values, then decreases, as shown in Fig. 14(a).

The peak value for the smaller $d/D$ occurs at $U_r=$ 5-7, respectively. However, it occurs at $U_r=$ 10 for $d/D=$ 1.5. Most of the values of $C_{L2,mean}$ are around zero Fig. 14(c), which is mainly induced by the symmetrical vibrations of the circular cylinders. It is worth noting that for $d/D=$ 1.5 and 2.0, the $X$-$Y$ trajectories of the rear circular cylinder at $U_r=$4 is not symmetrical about the central line, as shown in Figs. 11(b) and 12(b), which is responsible for the sudden “jumps” in the curves of $C_{L2,mean}$ vs $U_r$. Figs. 14(b) and (d) shows that both $U_r$ and $d/D$ have a great influence on the variations of the $C_{D2,rms}$ and $C_{L2,rms}$.

Fig. 13Time history of the force coefficients (CD, CL) and displacements (x/D, y/D) and power spectral density (PSD) functions of the displacement signals (x/D, y/D) at different cases

a)$d/D=$0.5, $U_r=$9

b)$d/D=$1.0, $U_r=$8

c)$d/D=$0.5, $U_r=$8

d)$d/D=$2.0, $U_r=$6

Fig. 14Hydrodynamic coefficients at different reduced velocities for the downstream 2-DOF circular cylinder

a) Mean drag coefficient, ($C_{D2,mean}$)

b) Root mean square of the drag coefficient, ($C_{D2,rms}$)

c) Mean lift coefficient, ($C_{L2,mean}$)

d) Root mean square of the lift coefficient, ($C_{L2,rms}$)

4.6. Vortex dynamics of the wake-induced vibration

To investigate the dynamics of the WIV system, two types of the wake developments are shown in Figs. 15 and 16 by means of the instantaneous vorticity contours as well as the motions of the circular cylinder within one cycle.

Fig. 15 shows the instantaneous vorticity changes at $d/D=$ 2.0 and $U_r=$ 14, with stages marked from (i) to (v). The time series of the circular cylinders inline and transverse motion curves and the $X$-$Y$ trajectories which are the one type of “dual-8” are also exhibited. Also in the $X$-$Y$ trajectory figures, the position of the circular cylinder and its moving direction are marked.

As started from stage (i) where the circular cylinder is in the upper position, the vortices are marked by “$A$-$B$-$C$-$D_1$-$D_2$-$E_1$-$E_2$-$F_1$-$F_2$-$G_1$-$G_2$”, which constantly move towards the downstream direction. From (i) to (ii), the circular cylinder moves downward and reaches the middle position; a new vortex “$D_3$” is generated from the tail of $D_1$. From (ii) to (iii), the circular cylinder keeps moving downward; “$D_3$” and “$B$” merge into a new vortex “$BD$”; a new vortex “$b$” is generated from the square cylinder. From (iii) to (iv), the circular cylinder moves upward, and arrives at the middle position again; vortex “$C$” develops into two vortices “$C_1$” and “$C_2$”; vortices “$G_1$” and “$G_2$” separate from each other. From (iv) to (v), the circular cylinder reaches the upper position; vortex “$BD$” develops into two vortices “$B{D}_1$” and “$B{D}_2$”; a new vortex “$b$” appears around the square cylinder. By comparing the starting stage (i), and the ending stage (v), it can be easily identified how the vortices are developing within one period and they correspond to: from “$A$-$B$-$C$-$D_1$-$D_2$-$E_1$-$E_2$-$F_1$-$F_2$-$G_1$-$G_2$”to “$a$-$b$-$A$-$B{D}_1$-$B{D}_2$-$C_1$-$C_2$-$D_1$-$D_2$-$E_1$-$E_2$”.

Fig. 15Instantaneous vorticity developments for the WIV system at Re= 120, d/D= 2.0 and Ur= 14

In Fig. 16 the flow pattern developments at $Re=$ 120, $d/D=$ 0.5 and $U_r=$ 9 are also investigated. At stage (i), the vortices are named as “$a$-$b$-$A$-$B$-$C_2$-$C_1$-$D_2$-$D_1$-$E$” while the circular cylinder is in the lower position. From stages (i) to (ii), as the circular cylinder moves upward, vortices “$B$” and “$D_2$” merge into a new vortex “$B{D}_2$”. In (iii), the circular cylinder is in the upper position, the vortices “$A$” and “$C_2$” merge into a new “$A{C}_2$”. From (iii) to (iv), the circular cylinder moves to the lower position again, and two new vortices “$c$” and “$d$” are marked. Therefore, the vortices relationship between (i) and (v) is: “$a$-$b$-$A$-$B$-$C_2$-$C_1$-$D_2$-$D_1$-$E$” in (i) and “$c$-$d$-$a$-$b$-$A{C}_2$-$B{D}_2$-$C_1$-$D$” in (iv).

Fig. 16Instantaneous vorticity developments for the WIV system at Re= 120, d/D= 0.5 and Ur= 9