Two-degree-of-freedom vortex-induced vibrations of two square cylinders in tandem arrangement at low Reynolds numbers

Abstract

Although flow past cylinders has a vast spectrum of applications in various fields of engineering, the existence of flow-induced vibrations (FIVs) add a need for a more precise and thorough approach when studying such flows. In this context, this work investigates the FIVs’ mechanism due to two square cylinders placed in tandem arrangement through a characteristics-based-split (CBS) finite element method. Both cylinders, with a low mass ratio ($m_r$ $=$ 2), are free to oscillate in in-line and transverse directions. The center-to-center distance of 5$D$ ($D$ being the cylinder side width) is kept fixed between the cylinders. The Reynolds number ($Re$) is ranged from 40 to 200 with an increment of 40, and computations are carried out for the reduced velocity ($U_r$) ranging from 3 to 13 under each $Re$ value. Results show that the effects of Re and $U_r$ on the dynamic responses are qualitatively similar for both cylinders. However, effects on the downstream cylinder are stronger due to the upstream wake. For $Re$ equal to or beyond 120, the in-line low-frequency characteristic becomes obvious at some higher values of $U_r$ for the upstream cylinder, whereas it exists over a wide range of $U_r$ for the downstream cylinder. For lower $Re$ (up to 120), cylinders show steady behavior at $U_r$ $=$ 5. However, for higher $Re$, they vibrate with the maximum amplitudes (especially the in-line amplitude of the downstream cylinder), and show complex wake flow at the same $U_r$. Compared to higher $Re$, the maximum vibration amplitudes in the transverse direction of lower $Re$ are relatively higher, while the opposite trend is observed for in-line vibration amplitudes. Several unstable trajectories are also identified other than the typical figure-eight trajectories. In context of vortex shedding, different permutation of steady state, 2S, P+S, and complex modes are observed for different values of $Re$. It is also noted that the vortex shedding pattern keeps changing in the far wake for higher $Re$.

Introduction

Vortex-induced vibrations (VIVs) is an important mechanism to consider when cylindrical structures are employed in fluid-flowing zones. The continuously increasing application of these structures is evident in various engineering disciplines, in particular, civil engineering (bridges, high-rise buildings), electrical engineering (transmission lines, high voltage towers) and marine engineering (riser tubes, underwater structures, offshore platforms). In this context, different researchers have provided significant insights into the various VIV aspects (Marris, 1964, Griffin and Ramberg, 1982, Parkinson, 1989, Sumer et al., 2006, Sarpkaya, 2004, Williamson and Govardhan, 2004, Williamson and Govardhan, 2008, Bearman, 2011, Bearman, 1984). Particularly, the additional stream-wise motion to the VIV of transversely oscillating cylinder was found to suppress the formation of a secondary vortex in an isolated cylinder case, which affects the phase of shedding and controls the sign of energy transfer (Jeon and Gharib, 2001). In the two-degree-of-freedom (2-DOF) VIV of a circular cylinder, Jauvtis and Williamson (2004) found that when the mass ratio ($m_r$) is greater than 6, VIV responses of the 2-DOF cylinder were similar to the one with only transverse VIV. Moreover, an additional branch called ‘super-upper’ branch appeared when $m_r$ values were lower than 6, where the “Super-upper” branch was associated with an increased amplitude of vibration in both in-line and transverse directions. VIV trajectories, experimentally obtained by Kang and Jia (2013), showed that the alteration in trajectories was due to the variation in the reduced velocities and the natural frequencies of vibration in in-line and cross-flow directions. Furthermore, they classified the in-line vibration as a type of multi-frequency vibration. Two sets of numerical simulations for a freely vibrating circular cylinder at $m_r$ $=$ 10 were conducted by Singh and Mittal (2005). In the first set, $Re$ (Reynolds number) was kept constant at 100 with variation in $U_r$ (reduced velocity), whereas $U_r$ was kept constant at 4.92 with variation in $Re$ in the second set. This was the first time P+S mode of vortex shedding was observed in free vibration.

The gap flow (flow passing between the cylinders) formed in the presence of two or more cylinders brings drastic changes to the flow characteristics (Zdravkovich, 1988, Jester and Kallinderis, 2004, Borazjani and Sotiropoulos, 2009, Wang et al., 2019). To examine the effect of an upstream stationary cylinder on the wake induced vibrations’ (WIVs) characteristics of transversely vibrating downstream cylinder, Carmo et al. (2011) conducted two and three-dimensional simulations at Reynolds numbers of 150 and 300, respectively. Significant changes in the displacement amplitudes and lock-in boundaries of the downstream cylinder were observed when compared to an isolated cylinder. In addition, for the range of $Re$ considered in the simulations, an insignificant effect of the three-dimensional wake structures on the dynamics of the structural responses was observed. Spacing ratios ($L∕D$) in the study of Carmo et al. were 1.5, 3, 5 and 8, however, results of $L∕D$ $=$ 1.5 and 5 were similar to those of $L∕D$ $=$ 3 and 8, respectively. Mittal and Kumar (2001) performed a numerical study of two freely vibrating circular cylinders in tandem and staggered configurations at $Re$ $=$ 100 and $m_r$ $=$ 2.7273. For all computations made with varying non-dimensional structural frequency, the behavior of the upstream cylinder was found to be similar to the case of an isolated cylinder. Due to the flow interference, the downstream cylinder placed in the wake of the upstream one showed wake flutter. Prasanth and Mittal (2009) conducted a study for similar cylinder configurations with same $Re$ values, where the value of mass ratio was 10 and $U_r$ was varied from 2 to 15. In this study, the upstream cylinder and the isolated cylinder were found to have qualitatively similar responses, except that the upstream cylinder oscillated with slightly larger peak amplitude. They also reported that the wake from the upstream cylinder was responsible for the unsteady flow in the downstream cylinder. Chung (2017) studied the 2-DOF VIVs of two circular cylinders at $Re$ $=$ 100 and $m_r$ $=$ 2 for twenty different arrangements of the cylinders. The author compared his results for a tandem arrangement with the earlier studies, having one or more parameters ($Re$, $m_r$) or DOF different from his study, and concluded that the change in these parameters or VIV-DOF could significantly alter the transverse vibration characteristics.

VIV and WIV of the vibrating square cylinder remain relatively less explored as compared to the circular cylinder case. Earlier studies (Bearman and Obasaju, 1982, Blevins, 1977, Chung and Kang, 2003, Yang et al., 2005, Jaiman et al., 2015, Sen and Mittal, 2015, Sen and Mittal, 2011) on non-circular (square or rectangular) cylinder reported that the change of cross-section from circular to non-circular altered some fundamental aspects of VIV. Zhao et al. (2013) carried out the simulations for three values of inclination angles (0°, $22.5^{\circ }$ and 45° with incoming flow) of a square cylinder at $Re$ $=$ 100. In their study, minimum vibration amplitude and narrowest lock-in region at the case of zero inclination were observed. Sen and Mittal (2015) studied the effects of $m_r$ on a square cylinder vibrating in both the in-line and transverse directions. In their study, galloping, which is an instability characteristic with low frequency and high oscillation amplitude, was absent for $m_r$ $=$ 1, while clear galloping was observed for the other higher values of $m_r$. The vortex shedding for the VIV regime ($m_r$ $=$ 1) was governed by the 2S and C(2S) modes, whereas for below and above the threshold vibration amplitude (0.7$D$) of the galloping regime, it was governed by the 2S and 2P+2S modes, respectively.

In the previous numerical and experimental studies, VIV problems of two square cylinders have been dealt with by restricting the motion of cylinders either in one or both directions. Besides, the effect of spacing between the cylinders has mostly been discussed. Sakamoto et al. (1987) experimentally examined the two important characteristics of VIV (i.e fluctuating aerodynamic forces and vortex shedding frequency) at $Re$ $=$ 27 600 and 55 200 for the two square cylinders arranged in tandems. In their study, the maximum values of mean drag and fluctuating forces along with the minimum Strouhal number were observed at the critical spacing ratio of 4. However, due to different flow patterns, these factors were reported to be drastically changing at other spacing ratios. A similar critical spacing ratio for a tandem cylinder pair was reported in an open tunnel test conducted by Luo and Teng (1990). It was also observed that the downstream cylinder was subjected to a negative drag pressure and the corresponding flow was of a reattachment type for $L∕D$ $\le$ 4. In contrast, the drag experienced by the downstream cylinder was positive and both cylinders shed fully developed vortices in their wake when $L∕D$ increased above 4. In a tandem arrangement of two square cylinders, Luo et al. (1999) encountered transverse galloping in the downstream cylinder when it was forced to oscillate transversely. For $Re$ $=$ 130, 150 and 500, Sohankar (2012) numerically studied the effects of cylinders’ spacing ($L$) ranging from 1.3$D$ to 13$D$ and distinguished three major flow regimes (single slender body regime, the reattachment regime and co-shedding regime), which depended upon the combination of $Re$ and $L$. Keeping the vibration amplitude ($y$/$D$) and $Re$ fixed at 0.4 and 100 respectively, Mithun and Tiwari (2014) performed a numerical investigation by varying the cylinders’ vibration frequency for two square cylinders vibrating in the transverse direction. The cylinder spacing ratio in their study ranged from 2 to 5. Vortex shedding was suppressed for spacing ratios of 3 and 4 for the stationary case, while clear vortices were observed from the downstream cylinder for all values of the spacing ratios in the vibrating case. Thus, shed vortices at excitation-to-vortex shedding frequency ratio ($f_r$) of 0.8 changed their patterns from 2S to 2P when spacing ratio changed from 3 to 4. This led to a sudden increase in mean drag and root mean square (RMS) lift coefficients. Chatterjee and Biswas (2015) carried out a numerical study for two rows of square cylinders arranged in a staggered fashion, where the wake dynamics of flow around the cylinders were analyzed for different values of transverse separation. The flow showed periodic nature when the transverse separations were larger, whereas it was more chaotic for smaller separations. Bhatt and Alam (2018) compared the numerical results of a transversely vibrating square cylinder, placed in the wake of a stationary cylinder at Reynolds numbers 100 and 200, to a stationary isolated square cylinder. Wake cylinder was placed at the stream-wise distances of 2$D$ and 6$D$ from the stationary cylinder. A significant role of gap flow in determining the vibration response of the wake cylinder was noted. To examine the effects of Reynolds number on the 2-DOF WIV characteristics of a square cylinder placed in the wake of stationary square cylinder, Han et al. (2018) performed simulations for $Re$ ranging from 40 to 200. It was observed that the resonance phenomenon changed from single to dual and vortex shedding modes within the lock-in regime changed from 2S to P+S and 2T when the Reynolds number was increased to 160 and 200. Moreover, it was found that the orbital trajectories changed from single figure-eight shape to complex- and dual-figure-eight shapes when $Re$ increased to 160 and 200.

Among the structural cross-sections used in the aforementioned engineering applications, studies on square cylinders have been scarce as compared to the circular ones. In earlier research, the studies on single square cylinder are limited to 2-DOF problems, whereas multiple square cylinders are limited either to stationary or to transverse vibration case. Furthermore, it can be concluded from the discussed research that the edge present in the square section, bluff bodies present on the upstream or the downstream and the low $m_r$ in case of 2-DOF are the leading causes for change in flow characteristics in VIV. However, their combined effect is still to be investigated. Motivated from the above discussion, the present work numerically investigates the dependence of VIV characteristics on $Re$ varying from 40 to 200 and $U_r$ ranging from 3 to 13 at each $Re$ for square cylinders in tandem with 2-DOF and low $m_r$ of 2. Additionally, we compare the key features of the downstream cylinder with that of our previous study (Han et al., 2018), where the upstream cylinder was stationary.

Various studies (Borazjani and Sotiropoulos, 2009, Papaioannou et al., 2006, Carmo and Meneghini, 2006) related to tandem cylinder problems have illustrated that three-dimensional effect remains weak up to $Re$ $=$ 200 and two-dimensional simulation is suggested to be accurate up to this threshold. In the recent years, Bhatt and Alam (2018), Han et al., 2014a, Han et al., 2018, Wang et al. (2014), and Ma et al. (2017) have successfully conducted two-dimensional simulations for various cylinder problems in tandem considering the range of Reynolds number up to 200. Therefore, the chosen Reynolds number range ($Re$ $=$ 40–200) in the current study limits the study within two dimensions, which enables us to discuss the effect of reduced velocity (3–13) for different values of $Re$ lying in the laminar flow regime.

Flow at low Reynolds number is basically associated with small characteristics length of the object or velocity. Such phenomena generally observe in biological, aerospace, chemical and geotechnical processing. Moreover, VIV characteristics observed at low Re are approximate predictor of VIV at high Re (Bao et al., 2012).

The remainder of this paper is as follows: A brief description of the numerical approaches to solve the fluid–structure interaction (FSI) problem is given in Section 2. Computational model is defined in Section 3. Results are presented and discussed in Section 4. In Section 5, we compared the downstream cylinder results with our previous study and some important conclusions are drawn in Section 6.