## 1. Introduction

Bridge hangers in suspension bridges are commonly deployed in pairs or groups with close spacing [

1], and the aerodynamic behavior of closely spaced cylinders becomes very complex and is considerably different from that of a single isolated one as a result of the interference effects. Severe wind-induced vibrations have been observed on several well-known suspension bridges including the Akashi-Kaikyo Bridge in Japan [

2], the Great Belt East Bridge in Denmark [

3], and the Xihoumen Bridge in China [

4]. Excessive wind-induced vibrations raise concerns about the fatigue life at both ends of the hangers and other structural members and can also cause visual discomfort for drivers.

The study of the aerodynamic performance of twin cylinders in cross flow is a topic of both fundamental and practical importance [

5,

6,

7]. The aerodynamic behavior of flexible cylinders as a result of wind actions can be dramatically altered by their proximity to neighboring structures [

8,

9,

10,

11]. Slender, identical, and parallel cylinders such as bundled overhead conductors, heat exchange tubers, hangers of suspension bridges, and chimney stack groups are particularly sensitive to wake interference effects, which are associated with fatigue damage or catastrophic failure [

12,

13]. Aerodynamic interference between two cylinders may induce flow separation, shear-layer development, reattachment, gap flow switching, vortex impingement, and quasi-periodic vortices, involving most of the generic flow features associated with various structures [

14]. Flow around twin cylinders provides an excellent model for gaining insight into the underlying flow physics around multiple structures and has attracted wide attention [

7,

15,

16,

17,

18,

19].

There are three arrangements for parallel twin cylinders: tandem arrangement, side-by-side arrangement and stagger arrangement. The effects of aerodynamic interactions on cylinders are significantly different for the three arrangements. For instance, the downstream cylinder is sucked by the upstream cylinder when twin cylinders are arranged in tandem closely, while it may suffer strong vibration when they are arranged in stagger with a certain relative position, which can lead to disastrous results on engineering structures. Previous studies have focused on multiple aerodynamic instabilities of the downstream cylinder such as vortex-induced vibration (VIV) [

20], wake induced galloping [

21,

22,

23,

24], and a combination of them [

25]. Instability may occur on both cylinders in a tandem arrangement with a small spacing (W/D < 3.2) at a low Reynolds number [

26]. Additionally, the vibration amplitude of one cylinder is sensitive to the spacing between the two cylinders and whether one of them is fixed or not. For the downstream cylinder, strong vibration results in particular cases for staggered arrangement [

27].

It is worth mentioning that flow-induced vibration can also occur on two rigidly coupled identical circular cylinders in tandem, staggered and side-by-side arrangements [

25,

28]. Zhao [

25] studied the synchronous vibration of coupled twin cylinders in the cross-flow direction at a low Reynolds number, and found that the gap between the two cylinders has a significant effect on the response. For a tandem arrangement, the lock-in regime of the reduced velocity for VIV is very different from that of a single cylinder; for a side-by-side arrangement, there is a specific phenomenon that the combination of VIV and galloping emerges at a certain range of spacing and reduced velocity. The VIV of two rigidly coupled circular cylinders of different diameters at low Reynolds numbers has also been investigated by researchers [

20]. Setting a small gap between the two cylinders is an effective measure to mitigate the vibration by reducing the vibration amplitude and narrowing the lock-in regime. Kim and Kim [

21] investigated the characteristics of wake galloping for two parallel/unparallel circular cylinders via wind tunnel tests at a Reynolds number of 2.0 × 10

^{5}. The unparallel disposition of two cylinders was effective to reduce wake galloping phenomena caused by unsynchronized motion along the cylinder with varying gap spacing. That study proposed a new method of vibration control for adjacent cylinders.

Flow around twin cylinders arranged in tandem is extremely complex due to the aerodynamic or hydrodynamic interaction, and the flow characteristics are of great practical importance for determining the flow-induced behaviors of cylinders [

29,

30]. Flow characteristics are mainly dependent on the characteristics (turbulence and incident angle) of incoming flow and the arrangement of cylinders. Researchers [

9,

31,

32] classified the relative position of the downstream cylinder as three typical regions: proximity interference region, wake interference region, and region of no-interference. Meanwhile, flow regimes of the three patterns have been comprehensively studied via numerical simulations and experiments in the last several decades [

10,

14]. Previous research indicated that critical spacing values for the three typical regions are mainly dependent on the turbulence and incidence angle of flow [

14]. When the Reynolds number is in the subcritical range, wake interference takes place in tandem and slightly staggered cylinders whose spacing exceeds a critical value of 3.5D–4.0D [

33]. In this case, the downstream cylinder wake is greatly affected by the upstream cylinder but not vice versa, thus the upstream cylinder is assumed to behave like a single isolated cylinder. The upstream cylinder flow is influenced slightly through a feedback mechanism by the downstream cylinder with the spacing at least up to 8D (with a reduced St and increased aerodynamic coefficients compared to those of an isolated cylinder) [

34,

35]. In addition, researchers [

36,

37] have discovered two completely distinct flow characteristics in laminar and turbulent regimes. Mizushima and Suehiro [

38] studied the flow around twin tandem cylinders by both direct numerical calculation and numerical simulation. The results showed there was a critical Reynolds number for the transition from steady symmetric flow to an oscillatory flow, and also a certain range of the gap spacing where physical quantities (such as aerodynamic coefficients and Strowhal number) show an abrupt change. Moreover, the transition of flow regime is also connected with the flow incidence angle in the high subcritical range of the Reynolds number, and the critical incidence angle of transition is dependent on the Reynolds number and spacing of the cylinders [

37].

Pressure measurement is a familiar efficient method for the study of structural wind engineering and has been widely used in previous investigations on the aerodynamic characteristics of twin cylinders [

39,

40,

41]. As an essential aerodynamic parameter, the distribution and pulsation of pressure are sensitive to the surface roughness [

42], Reynolds number, flow incidence angle, diameter ratio [

43], and arrangement of the cylinders [

14,

44]. Igarashi [

45] tested the pressure on two tandem identical cylinders at a Reynolds number range of subcritical values. Vortex shedding was detected from the upstream cylinder in a certain spacing near 3.5D, with a sharp peak of pressure pulsation emerging at θ = 40° (where θ is the angle on the circumference taken from the front stagnation point on a cylinder) of the downstream cylinder. This location and magnitude remained unchanged up to the spacing of 7D, and the vortex-shedding frequency increased with rising flow velocity. Similar conclusions have been found by Arie et al. [

46]: the RMS (Root-Mean-Square) surface pressure was much higher for the downstream cylinder than for the upstream cylinder at the spacing of 4D and the Reynolds number of 1.57 × 10

^{5}, and a distinct peak of RMS pressure was observed at θ = 50° of the downstream cylinder. In addition, the RMS lift and drag of the cylinders were heavily dependent on the spacing, and were much larger for the downstream cylinder than for the upstream cylinder at the spacing ranging from 2D to 7D. Researchers [

47] also paid many efforts to study the effects of flow incidence angle on the flow characteristics and aerodynamic characteristics of parallel twin cylinders. Gu et al. [

48] classified three different pressure distribution patterns on the downstream cylinder and observed two switching processes for the wind incidence angle varying from 0° (in tandem) to 90° (in side-by-side) at high subcritical Reynolds numbers. A high level of asymmetric distribution of fluctuating pressure was detected at a high subcritical Reynolds number. The pressure pattern around the downstream cylinder may switch between two patterns at the critical wind incidence angle. This switching will create a step change of lift force on the downstream cylinder. However, for a supercritical Reynolds number, things will change. Effects of aerodynamic interference on the fluctuating pressures of twin circular cylinders of various wind incidence angles become weaker at the supercritical Reynolds number than those at the subcritical one [

27]. Furthermore, mean pressures on downstream cylinders at the supercritical Reynolds number have very different features from those at the subcritical one.

Previous studies were mainly devoted to studying the aerodynamic characteristics and the ensuing vibrations of two twin cylinders that are not coupled. In other words, the two cylinders respond independently to wind and wake excitations. However, serious synchronous vibrations of twin coupled cylinders still exist in several engineering structures, such as bundled conductors [

5], linked buildings [

49], twin hangers or cables in bridges [

28]. Unfortunately, to the authors’ knowledge, the vibration mechanism for coupled twin cylinders seems to be less understood. In the previous study by the authors [

28], it is shown that the coupled twin cylinders suffered from wake-induced vibrations at certain spacings and ranges of wind attack angles. The purpose of this work is to describe the aerodynamic forcing characteristics and flow patterns of coupled parallel twin circular cylinders through wind tunnel tests and numerical simulations. Pressure measurements were performed on a stationary section model with a series of cylinder spacings and wind incidence angles. Based on the pressure data, mean and fluctuating wind pressure distributions, reduced frequencies and aerodynamic forces of the cylinders are obtained and analyzed. In addition, numerical simulations were performed for flow visualization. Moreover, the exciting mechanism of synchronous vibrations for the coupled twin cylinders is also discussed from the point of view of aerodynamic forces.

## 4. Flow Visualization

In order to better understand the flow characteristics and wake interference between two cylinders, two-dimensional fluid computations using CFD (computational fluid dynamics) technology were also performed to visualize the flow regimes around cylinders. RANS (Reynolds Averaged Navier–Stokes) and LES (Large Eddy Simulation) are two most popular methods in CFD simulation. In recent decades, they have been widely used in the simulation of flow around cylinders, and the efficiency of them has been verified by many previous studies [

29]. By contrast, RANS is much less computationally intensive than LES and is sufficient to describe the nature of the problem in this study. Therefore, RANS equations were chosen and solved with the SST k-ω turbulence model in this study [

8,

51]. Before formal simulation, a mesh convergence test for the numerical model was carried out on a single circular cylinder (as shown in

Appendix A). The computational domain used for the numerical model with a height of 45D and width of 68D was discretized by about 250,000 cells, and the mesh near cylinders is shown in

Figure 11. Furthermore, to validate both the present experimental and numerical method, the results of a single circular cylinder are compared with the data of other literatures in

Table 1. It can be seen that the present results are in agreement with the literature to a certain extent, especially those of the experiment. Nevertheless, the drag coefficient by numerical simulation of this paper is about 16~25% smaller than those of the experiments, while the corresponding numerical result of [

52] is about 13~27% smaller than the experiment results. Present Strouhal numbers (St) also agree well with those of the previous papers. To sum up, errors in this paper can be generally accepted.

According to the discussion in

Section 3, there are two typical flow patterns for twin cylinders both in tandem and staggered arrangements with various spacings. Therefore, two typical spacings of 2.5D and 4.3D are selected for numerical computation.

Figure 12 shows the velocity contours for twin cylinders arranged in tandem. It can be seen that the flow regime around the twin cylinders shows two typical patterns. For W/D = 2.5, both shear layers from the upstream cylinder symmetrically reattached to the downstream one, and as outlined above, flow velocity is very low in the gap between cylinders. Meanwhile, the vortex formation of the downstream cylinder is not identifiable. In the other flow pattern, for W/D = 4.3, small vortices were formed in the space between the cylinders and convected toward the downstream cylinder. In this case, evident vortices are formed in the wake of the downstream cylinder, indicating that the interference effect of the upstream cylinder is weak when compared with the other pattern.

When twin cylinders are placed in a staggered arrangement, things become different. It is worth mentioning that in numerical simulation, the total lift coefficient of the twin cylinders at W/D = 2.5 comes to a peak at α = 10°, differing from the experiment result that the total lift coefficient comes to a peak at α = 9°. This difference may be caused by the large gradient of 3° for the wind incidence angle in the wind tunnel test. To make better sense of the distinct flow pattern at the critical angle, the velocity contours for twin cylinders arranged in a staggered arrangement at α = 10° are shown (

Figure 13) and discussed in this paper. At W/D = 2.5, the vortex formation of the downstream cylinder is identifiable, while the vortex shedding of the upstream cylinder is completely suppressed by the downstream one. In this case, as discussed in

Section 3, the wake flow of the upstream cylinder flows rapidly near the top edge and impacts on the inlet edge of the downstream cylinder, which leads to a positive lift and drag, respectively. When W/D = 4.3, the interference becomes weaker and apparent vortexes are generated from both the twin cylinders. Nevertheless, the vortex street is asymmetrical on account of the staggered arrangement. In general, the flow visualization agrees well with the results of pressure measurement in this paper. In addition to the flow-induced vibrations of twin cylinders, the conclusions drawn above could also be useful for evaluating the wind pressures on structures with multiple cylinders [

54,

55,

56].

## 5. Conclusions

The aerodynamic forcing characteristics of coupled twin circular cylinders were investigated experimentally to study the synchronous galloping. Pressure measurements were performed on a stationary section model with a series of cylinder spacings and wind incidence angles. Based on the pressure data, mean and fluctuating pressure distributions, reduced frequencies and total aerodynamic forces of the cylinders are discussed. Moreover, numerical simulations were performed for flow visualization. The conclusions are summarized as follows:

(1) The flow around twin cylinders shows two distinct patterns with various spacings, and the corresponding critical spacings at wind incidence angles of 0° and 9° are in the range of 3.8D~4.3D and 3.5D~3.8D, respectively.

(2) When W/D is greater than the critical value, regular vortex shedding will be generated from the upstream cylinder and the vortexes impact upon the downstream cylinder. In consequence, large fluctuating pressure on the downstream cylinder is induced, while the fluctuation decreases with increasing W/D. When W/D is smaller than the critical value, the vortex shedding of the upstream cylinder is suppressed by the downstream cylinder: the symmetric region between the twin cylinders is almost static at a wind incidence angle of 0°; a negative pressure region emerges on the top edge of the downstream cylinder and the pressure distribution is asymmetrical at a wind incidence angle of 9°.

(3) When 2.5 ≤ W/D ≤ 3.5, there is a critical wind incidence angle near 9°. In this case, the wake flow of the upstream cylinder flows rapidly near the top edge and impacts on the inlet edge of the downstream cylinder, which causes a negative and positive pressure region near the corresponding edge regions, respectively. Hence, the lift force of the downstream cylinder comes to a peak while the drag force jumps to a higher value. In consequence, the total lift force of the twin cylinders comes to a peak while the total drag force jumps to a higher value. Furthermore, there is a sharp drop in the total lift coefficient in the range of 9° < α < 12°, and this may lead to aeroelastic instability of the coupled twin cylinders when the flow velocity increases to a critical value. Therefore, these cases (i.e., 9° < α < 12° while 2.5 ≤ W/D ≤ 3.5) should be carefully checked in engineering applications involving avoiding aeroelastic instability.