## Abstract

This paper numerically investigates the flow-induced vibration of a circular cylinder attached with front and/or rear splitter plates at a low Reynolds number of Re = 120. The effects of plate length and plate location on the hydrodynamic coefficient, vibration response, and flow wake are examined and discussed in detail. The results reveal that the hydrodynamic coefficient of the cylinder with a single rear plate is significantly reduced at Ur ≤ 8 (Ur is the reduced velocity), resulting in the vortex-induced vibration (VIV) suppression. Nevertheless, the galloping is excited at Ur > 8 due to the hydrodynamic instability, accompanied by the jump of response amplitude and hydrodynamic force, as well as the abrupt drop of response frequency. The alternate reattachment of shear layers on the plate surface introduces an extra lift force that strengthens the vibration response. By introducing an individual front plate, significant VIV suppression is achieved. The vibration exhibits variable patterns when the cylinder is equipped with bilateral plates, including the typical VIV mode, weak VIV-galloping coupling mode, and IB-galloping-DB mode (IB and DB represent the initial branch and desynchronization branch of VIV, respectively). The galloping branch in IB-galloping-DB mode is observed with an abrupt drop in response frequency, as well as a tiny time lag between the displacement and lift force. The vibration response is significantly suppressed when the cylinder is simultaneously equipped with a 1D front plate and a 1–2D rear plate due to the streamlined profile.

## 1 Introduction

In nature and many engineering applications, fluid flow past a bluff body can be observed widely, particularly for the circular cylinder. It is a classical problem in fluid mechanics and has been widely investigated over the past few decades [15]. When the flow behind a bluff body becomes unsteady, a well-known phenomenon of alternate shedding vortices and oscillating dynamic loads occurs. Moreover, the bluff body may vibrate if it is elastically mounted or free to move. When the vortex shedding frequency is close to the structural nature frequency, more severe vibrations can be induced, which is usually considered as the origin of structural fatigue failure. Therefore, a lot of studies have been conducted on the topic of suppression of vortex shedding, mainly in terms of the principle of altering the wake and vortex generating conditions to reduce the dynamic forces. A suppression device affixed to a bluff body is one of the common approaches. As reviewed by Choi et al. [6], Owen et al. [7], Rashidi et al. [8], and Zdravkovich [9], some passive add-on devices have been widely proposed and put into practice, including splitter plates, strakes, fairings, small rods, etc.

Since Roshko’s experiment in 1954, placing a splitter plate behind a circular cylinder has been proved to be an effective passive method to reduce drag force [1012]. The drag-reduction mechanism is attributed to that the splitter plate can prevent the separated shear layers near the body from rolling up [13], and consequently lead to a substantial recovery of the base pressure in the near wake [11,14]. Plentiful evidence has shown that the Reynolds number (Re = UD/υ, where U is the freestream velocity, D is the diameter of a circular cylinder, and υ is the fluid kinematic viscosity) and plate length (L) can significantly influence the vortex shedding behavior and hydrodynamic forces. Even with a very short splitter plate of L/D = 1/16, a 9% drag-reduction could be achieved as reported in Apelt et al. [11]. The maximum drag-reduction is achieved when the plate length is approximately equal to the vortex formation length [15], which is dependent on Re [16]. In general, to achieve a maximum drag-reduction, the introduced splitter plate is usually shorter at high Re than that at low Re. For example, Apelt et al. [11] reported the optimum plate length of L/D = 1 at l04 < Re < 5 × l04, but the results of Kwon and Choi [15] recommended a larger length of L/D = 4.5 at Re ≤ 160.

Although a splitter plate can postpone the interaction of shear layers and thus reduce the hydrodynamic forces, it cannot completely suppress the flow-induced vibration (FIV), as many scholars have pointed out [13,1719]. Vortex-induced vibration (VIV) and galloping are the most common patterns of FIV. Various papers have discussed these two vibrations [2023], as well as their interaction [2426]. It is well-known that a circular cylinder can be vortex-excited but cannot gallop. However, the cylinder is more likely to gallop in the presence of a rigid splitter plate. Assi et al. [27] have claimed that a circular cylinder experiences a galloping response when it is attached to a 0.25–2.0D rigid splitter plate. Moreover, the visualization of the flow field around the structure revealed that the reattachment of shear layers on the tip of plates is the hydrodynamic mechanism that drives this excitation [28]. Stappenbelt [29] conducted experiments in a water tunnel about a circular cylinder attached with rigid splitter plates. Results showed that the galloping-type response halts abruptly at high reduced velocity ranges when the plate is shorter than 0.5D. In the range of 0.5 < L/D < 2.8, a no-limited galloping response is presented. Further lengthening plate, no significant VIV or galloping is observed. Liang et al. [30] studied the FIV response of a circular cylinder with a rigid splitter plate placed downstream of a cylinder. Four responses for different plate lengths are identified: (1) VIV-only at L/D = 0.4, 0.5; (2) full interaction between VIV and galloping at L/D = 1.0, 1.5; (3) a VIV regime and partial interaction between VIV and galloping at L/D = 2.0, 2.5, 3.5; and (4) VIV regime and classical galloping at L/D = 4.5, 5.0.

Compared to the considerable research on a rear splitter plate, studies on the evaluation of a front splitter plate or bilateral plates arranged on both sides of a cylinder are fewer. Gao et al. [31] reported a drag-reduction of 36% at the optimal plate length of L/D = 1.0 when a splitter plate is implemented onto the upstream surface of a circular cylinder. Their results supported by particle image velocimetry measurement suggested that the front plate can modify the wake-flow and hydrodynamic forces by changing incoming flow conditions. This drag-reduction can be further attributed to a flow separation delay phenomenon, as Chutkey et al. [32] reported that a front splitter plate with a 1D length can delay the separation point of the boundary layer from 82 deg to 122 deg. Qiu et al. [33] compared the effect of plate location on the drag forces. Results indicated that there exists a critical Reynolds number range (Rec) of 3.0 × 105–3.31 × 105 to achieve drag-reduction. The arrangements of a front plate or bilateral plates perform better at Re < Rec. However, the opposite affection occurs when Reynolds numbers are beyond the critical value.

In our earlier study [17], the flow-induced vibration response of a circular cylinder with splitter plates placed upstream and downstream individually and simultaneously was numerically investigated. A small gap of G/D = 0.2 was considered, where G is the streamwise distance from the rear base point of the cylinder to the leading edge of the splitter plate. By introducing a rear splitter plate, the reattachment of shear layers on the plate surface can generate an extra lift force and thus contributes to galloping oscillation. Interestingly, the galloping can be significantly suppressed by placing a front plate or bilateral plates. It should be noticed that the lengths of these two splitter plates are identical when considering a bilateral arrangement. Therefore, to further improve our studies on the VIV and galloping response of a circular cylinder in the presence of splitter plates, a new numerical investigation is conducted in this paper, where the gap between cylinder and plate is absent and the effect of bilateral plates with different lengths is examined.

## 2 Problem Description

Flow-induced vibration of the present cylinder-plate body is modeled by a mass-spring-damping system and numerically investigated at a low Reynolds number of 120. The structure can move both in the in-line and cross-flow directions. As Fig. 1(b) displays, that the diameter of the circular cylinder is D. The front and rear splitter plate lengths are defined as Lu and Ld, respectively. The plate thickness is set as δ = 0.1D. These plates are affixed to the cylinder with no gap. The mass ratio m* of the present cylinder-plate body is 6.9, where m* = m/md, and m, md represent the structure mass and the displaced mass of fluid, respectively. The damping ratio ζ is set as 0.01, where $ζ=C/(2K(m+ma))$, and ma, C, and K are the added mass, system damping, and spring stiffness, respectively. Therefore, a low m*ζ of 0.069 is adopted in this work with the intent of exciting a considerable high-amplitude response. To cover a wide reduced velocity range of Ur = 3–18, the natural frequency fn of the present cylinder-plate body in still water is set as a variable by varying the system damping accordingly, expressed as
$fn=12πKm+ma$
(1)
where ma = Camd, Ca is the added mass coefficient and one can see more detailed information and calculating method about Ca in the report by Vikestad et al. [34]. As a result, the variation of Ur is achieved without the change of Re. The main simulation parameters in this work are listed in Table 1.
Fig. 1
Fig. 1
Close modal
Table 1

Simulation parameters

ParameterSymbolValue
Mass ratiom*6.9
Damping ratioζ0.01
Length of the front splitter plateLu/D0.5, 1
Length of the rear splitter plateLd/D0.5, 1, 1.5, 2
Thickness of the splitter plateδ/D0.1
Reduced velocityUr3–18
Reynolds numberRe120
ParameterSymbolValue
Mass ratiom*6.9
Damping ratioζ0.01
Length of the front splitter plateLu/D0.5, 1
Length of the rear splitter plateLd/D0.5, 1, 1.5, 2
Thickness of the splitter plateδ/D0.1
Reduced velocityUr3–18
Reynolds numberRe120

As displayed in Fig. 1(a), the computational domain is a rectangular region with a size of 70D (in-line direction) × 40D (cross-flow direction), yielding a blockage ratio of 2.5%, which is defined as the ratio of the structure’s projected length with respect to the incoming flow to the width of the computational domain. The circular cylinder’s center locates at a distance of 20D from the upstream and two lateral boundaries. For the upstream boundary, a uniform velocity profile is specified as u = uin and v = 0, where u and v are the velocity components in the x and y directions, respectively. The outlet boundary adapts the zero gradient condition for the flow velocities along the two directions (∂u/∂x = 0 and ∂v/∂x = 0). No-slip condition is applied to the velocity on the structure surface. The normal component of velocity and the tangential component of stress vectors are prescribed as zero on the lateral symmetry boundaries.

To improve the efficiency of mesh-updating and capture flow characteristics, the computational domain is divided into three zones: accompanying moving zone, dynamic mesh zone, and static mesh zone. As shown in Fig. 1(a), a circle of diameter 6D concentric with the circular cylinder, combined with a rectangle region of 6D × 4D, is defined as an accompanying moving zone, which follows the structure’s movement. A slightly larger rectangle region (25D × 20D) enveloping the accompanying moving zone is set as the dynamic mesh zone, where the grids split or merge in each time-step. The rest of the computational domain is a static mesh zone, where the grids are unchanged to save computational cost.

## 3 Mathematical Method

### 3.1 Governing Equations.

Fluid flow past present cylinder-plate body is modeled by the two-dimensional unsteady incompressible Navier–Stokes (N–S) equations, including the continuity and momentum equations. In order to fix the fluid-structure interaction, an arbitrary Lagrangian–Eulerian (ALE) reference frame is employed as follows:
$∂ui∂xi=0$
(2)
$∂ui∂t+(uj−ujm)∂ui∂xj=−1ρ∂p∂xi+υ∂2ui∂xj∂xj$
(3)
where xi is the Cartesian coordinate in i direction, ui is the velocity component in the direction xi, ujm is the advection velocity due to the mesh deformation in the ALE method, t is flow time, p is pressure, and ρ is the fluid density.

Finite volume method and pressure implicit with the splitting of operators algorithm are employed for the pressure-velocity coupling. The spatial discretization of the convective term is performed by the fourth-order cubic scheme, while the diffusion term is carried out by the second-order linear scheme. The time derivative term is discretized using a blended scheme consisting of the second-order Crank–Nicolson scheme and a first-order Euler implicit scheme [35].

The flow-induced vibration responses of the present cylinder-plate body in the in-line and cross-flow directions can be expressed as [3638]
$(m+ma)x″+Cx′+Kx=fD(t)$
(4)
$(m+ma)y″+Cy′+Ky=fL(t)$
(5)
where x″, x′, and x represent the structure acceleration, velocity, and displacement in the in-line direction, respectively, and y″, y′, and y represent the same parameters associated with the cross-flow motion, fD(t) and fL(t) present the drag and lift force, respectively.

To solve the fluid-structure interaction, the ansys-fluent package was employed with the help of user-defined functions. The sequential fluid-structure interaction iteration was adopted with the initial condition of x′ = 0, x = 0, y′ = 0, and y = 0. In each time-step, the flow field is firstly obtained by solving N–S equations. After that, the hydrodynamic forces are calculated by conducting an integration involving the pressure and viscous stresses. Finally, the displacements are computed by substituting the hydrodynamic forces into Eqs. (4) and (5), which are discretized by an improved fourth-order Runge–Kutta method [39,40]:

$y(tn+1)=y(tn)+y′(tn)Δt+Δt26(S1+S2+2S3+S4)$
(6)
where,
$S1=fL(tn)m+ma−Cm+may′(tn)−Km+may(tn)$
(7)
$S2=fL(tn)m+ma−Cm+ma[y′(tn)+Δt2S1]−Km+ma[y(tn)+Δt2y′(tn)]$
(8)
$S3=fL(tn)m+ma−Cm+ma[y′(tn)+Δt2S2]−Km+ma[y(tn)+Δt2y′(tn)+Δt28S1]$
(9)
$S4=fL(tn)m+ma−Cm+ma[y′(tn)+S3Δt]−Km+ma[y(tn)+y′(tn)Δt+Δt22S1]$
(10)
where S1, S2, S3, and S4 are the intermediate functions, Δt is the normalized time-step, which is set as 0.001 to meet the demand that the maximum Courant–Friedrichs–Lewy number in the computational domain is below 0.5. After obtaining the displacement data, the mesh is accordingly updated so that a new mesh is prepared for the flow field calculation in the next time-step. To ensure reliable numerical results, the residual less than 10−5 was selected as the convergent criteria. The calculation stopped when sufficient periodic results (more than 100 cycles) were obtained for statistical analysis.

### 3.2 Computational Mesh and Mesh Dependence Check.

Figure 2 shows the computational mesh used for the present simulation. As mentioned earlier, the computational domain includes three zones. Static and structured quadrilateral cells are used in zone I. The region near the structure surface is also equipped with structured quadrilateral cells to make sure that the flow in boundary layer grids owns a relatively high computational resolution, while the rest part in the accompanying zone (zone III) is tessellated with triangular elements. The mesh in zone III varies synchronously with the structure motions. The deforming zone (zone II) is also filled with unstructured triangular elements, which can be adjusted and updated based on structure motion.

Fig. 2
Fig. 2
Close modal

The mesh independence study is conducted first, before proceeding to the normal simulation. The results of four mesh systems with different resolutions are compared in Table 2. The percentage difference (absolute value) is calculated by using the results of M4 as a reference. It is clearly seen that the variations of the normalized in-line and cross-flow amplitudes (AX* = AX/D and AY* = AY/D, where AX and AY are the in-line and cross-flow amplitudes, respectively) converge at M3. For further refinement from M3 to M4, the difference is less than 0.5%, indicating the variation can be negligible. Therefore, M3 is adopted. With this mesh resolution, the perimeter of the circular cylinder and the plate thickness are divided into 360 and 16 segments, respectively. The height of the first mesh layer next to the structure is 0.0004D, corresponding to the requirement of y+ < 1.

Table 2

Mesh independence study on a circular cylinder with a 1D rear splitter plate at Ur = 12

MeshNodes on the surface of the cylinder-plate bodyElementsAY*AX*
M130050,5820.82679 (8.61%)0.03235 (6.29%)
M234681,7300.8826 (2.45%)0.03397 (1.59%)
M3392113,3280.90122 (0.39%)0.03443 (0.26%)
M4434145,8080.904730.03452
MeshNodes on the surface of the cylinder-plate bodyElementsAY*AX*
M130050,5820.82679 (8.61%)0.03235 (6.29%)
M234681,7300.8826 (2.45%)0.03397 (1.59%)
M3392113,3280.90122 (0.39%)0.03443 (0.26%)
M4434145,8080.904730.03452

### 3.3 Model Validation.

The numerical model is first validated against a benchmark case of flow past a stationary cylinder-plate body at Re = 100, where the splitter plate with 1D length is attached to the rear base of the cylinder. The validations are made for the time-averaged coefficient (CD,mean), Strouhal number (St), and the normalized length of the recirculation bubble (Lr* = Lr/D, where Lr represents the streamwise distance from the center of the circular cylinder to the end of recirculation bubble). Table 3 shows the details of the comparisons. The maximum differences of St, CD,mean, and Lr* are observed to be 5.87%, 3.65%, and 5.36%, respectively, showing a relatively good agreement between previous and present results. After that, the numerical model is further validated with reported data by Bao et al. [41], Wu et al. [42], and Borazjan and Sotiropoulos [43] for flow-induced vibration response of a bare circular cylinder. Figure 3 illustrates a good coincidence in the normalized cross-flow amplitude. The upper branch observed by Williamson and Govardhan [44] and Jauvtis and Williamson [45] disappeared at this low Re due to the greater viscous force. Nevertheless, the initial branch (IB), the lower branch (LB), and the desynchronization branch (DB) are still observed. Generally, the present mathematical model and method could precisely predict the structural vibration.

Fig. 3
Fig. 3
Close modal
Table 3

Comparisons of time-averaged drag coefficient (CD,mean), Strouhal number (St), and normalized length of recirculation bubble (Lr*) for a stationary circular cylinder attached with a splitter plate on its rear base with L/D = 1 at Re = 100

ReferencesCD,meanStLr*
Kwon and Choi [15]1.1800.1373.71
Lu et al. [47]1.1680.137
Hwang et al. [48]1.1700.1373.71
Deep et al. [49]1.2440.142
Present result1.1750.1373.92
Maximum difference5.87%3.65%5.36%
ReferencesCD,meanStLr*
Kwon and Choi [15]1.1800.1373.71
Lu et al. [47]1.1680.137
Hwang et al. [48]1.1700.1373.71
Deep et al. [49]1.2440.142
Present result1.1750.1373.92
Maximum difference5.87%3.65%5.36%

## 4 Results and Discussion

### 4.1 Effect of Plate Length

#### 4.1.1 Hydrodynamic Coefficient.

The hydrodynamic coefficient of a cylinder with a rear splitter plate of Ld/D = 0.5, 1, 1.5, and 2 against the bare cylinder are plotted in Fig. 4. The time-averaged drag coefficient (CD,mean) and the root-mean-squared lift coefficient (CL,rms) are calculated by
$CD,mean=1N∑i=1N2fD(t)ρuin2D$
(11)
$CL,rms=1N∑i=1N[2fL(t)ρuin2D]2$
(12)
where N is the total number in the time series for statistics. Agreeing well with the results of Wu et al. [42] and Bao et al. [41], both CD,mean and CL,rms of the bare cylinder increase first as Ur increases from 3, and they peak at Ur = 6 and 5, respectively, followed by a drop and then remain a relatively stable value at 8 ≤ Ur ≤ 18. From the CD,meanUr and CL,rmsUr curves, it is clearly seen that there exist an obvious difference in hydrodynamic coefficient between the cylinder-plate body and bare cylinder. As illustrated in our earlier work [17], with the introduction of a rear splitter plate of 1D, 1.5D, and 2D, the drag forces are completely suppressed in the whole Ur range in comparison with the bare cylinder. However, in this work, a drag-increase phenomenon is observed for a shorter plate of Ld/D = 0.5 at 9 ≤ Ur ≤ 18. This was also reported by Stappenbelt [29] and Achenbach [50], where the drag forces of the cylinder-plate body are larger than that of a bare cylinder in the high reduced velocity range when the plate is shorter than 0.5D. Nevertheless, the drag forces of the cylinder-plate body decrease as plate length increases from 0.5D, which is due mainly to the improved streamlining of the structure.
Fig. 4
Fig. 4
Close modal

Compared to the bare cylinder, the root-mean-squared lift coefficient is obviously diminished at Ur ≤ 8, 10, 10, and 11, when the cylinder is equipped with a splitter plate of Ld/D = 0.5, 1, 1.5, and 2, respectively. These inflection points are in one-to-one correspondence to those of drag coefficients. Nevertheless, the local peaks are still observed at Ur = 5 for all considered cases, presenting a typical VIV response [28]. After a relatively stable stage, CL,rms of the cylinder-plate body abruptly rise around Ur = 8–11, and then maintain a large value. As Assi et al. [27], Assi and Bearman [28], and Liang et al. [30] suggested, the sharp rise of CL,rms is attributed to the hydrodynamic instability which usually causes the galloping response. Besides, the onsets of hydrodynamic instability and variations of CL,rms at galloping range are closely related to plate length. For a 0.5D rear plate, the rapid rise of CL,rms occurs at Ur = 8–9, after that a monotonously linear decline is observed. The onset of hydrodynamic instability emerges at Ur = 10–11 both for 1D and 1.5D rear plate. However, the deceleration of CL,rms for 1.5D plate is much slower than that of 1D plate. Further increasing the plate length to 2D, the abrupt rise of CL,rms occurs at a higher Ur range between 11 and 12. Meanwhile, the variations become relatively stable at Ur = 13–18, indicating that reduced velocity is less influential. In general, the shorter plates own the smaller onset Ur of hydrodynamic instability, and the larger decline of CL,rms at a high Ur range. As Akilli et al. [51] and Kwon and Choi [15] reported, the flow field around the cylinder-plate body can be hugely modified when the plate length is larger than 1D. However, the wake flow is more unstable at Ld/D ≤ 1. Therefore, the greater reduction of lift coefficients for the cases of Ld/D = 0.5 and 1 is observed in the galloping regime, as compared to those of Ld/D = 1.5 and 2. In contrast, there are no obvious fluctuations of drag coefficients with increasing the reduced velocity when the structure enters the gallop region, as those reported by Stappenbelt [29] and Assi and Bearman [28].

The phase portraits of drag and lift coefficients are compared in Fig. 5. The repeated motion trajectories indicate that the fluctuations of these two coefficients reach a periodic but statistically stable state. It is clearly seen that the Lissajous traces present regular or irregular figure-eight orbits, presenting that the drag frequency is approximately as twice as the lift frequency. The regular figure-eight and C-shape orbits are responsible for the phase difference between the drag and lift coefficients.

Fig. 5
Fig. 5
Close modal
The distributions of time-averaged pressure coefficients around the structure at four typical reduced velocities are compared in Fig. 6. The time-averaged pressure coefficient is calculated by
$CP,mean=1N∑i=1Np−p∞0.5ρuin2$
(13)
where p is the absolute pressure and p is the reference pressure at the inlet boundary. For a bare cylinder, it is found that the low-pressure zone at Ur = 6 is more prominent, yielding a larger pressure difference and therefore a larger drag force, as compared with other circumstances. By introducing a rear plate, the local low-pressure zone is significantly reduced, explaining the drag reduction. However, the low-pressure zones for Ld/D = 0.5 at Ur = 9 and 18 are wider in the cross-flow direction, leading to a drag increase compared to that of a bare cylinder. As the plate increases from 0.5D to 2D, the absolute values of negative pressure are decreased with the accompaniment of a shrinking local low-pressure zone. Consequently, the 2D plate performs best in drag-reduction as shown in Fig. 4. For the cylinder-plate body, hydrodynamic instability is observed at high reduced velocities as the low-pressure zone is enlarged. The low-pressure zone partially or completely enveloping the plate contributes to the lift force acting on the top and bottom sides of the plate surface, which should be also taken into account for the total lift. Therefore, the enlargement of the low-pressure region is possibly responsible for the increase of lift force.
Fig. 6
Fig. 6
Close modal

#### 4.1.2 Vibration Amplitude and Frequency Response.

Figure 7 shows the structure’s amplitude responses in both in-line and cross-flow directions. The associated results of a bare cylinder are also illustrated for comparison. It is seen that the vibration responses of the cylinder-plate body are significantly different from that of a bare cylinder. For a bare cylinder, the amplitude increases at Ur = 3–6, then declines at Ur = 6–8, and finally keeps relatively stable at Ur = 8–18, consequently named initial, lower, and desynchronization branches. It is noted that the peak amplitude occurs at Ur = 6, larger than that of lift coefficient at Ur = 5. This is because the cylinder needs to absorb adequate energy before it reaches a large amplitude [41]. For the cylinder with a rear plate, the vibration responses are remarkably suppressed as compared with the bare cylinder in the initial and lower branches, where the maximum amplitude-reductions reach 98.56% and 87.76% in the in-line and cross-flow directions, respectively. However, the amplitude rises sharply at Ur = 9–18 due to the hydrodynamic instability. This rapid growth in amplitude shows a galloping regime [4,52,53]. The onset reduced velocities of galloping regime are 8, 10, 10, and 11 for 0.5D, 1D, 1.5D, and 2D plate, respectively, corresponding to variations of lift coefficient in Fig. 4. Generally, in the galloping regime, the longer the plate, the smaller the cross-flow amplitude, but the larger the in-line amplitude. These findings illustrate that the introduced rear splitter plate can significantly reduce the response amplitudes in the VIV region, while a huge enhancement is observed at a larger Ur range where galloping is excited. Compared with our earlier reported results [17], the main findings include: (1) the cylinder-plate body could gallop when the reduced velocity is sufficiently high, which is not sensitive to the gap; and (2) the structure vibration is suppressed at the gap ratio of G/D = 0.2. It indicates that a small gap does not enhance the communication between two shear layers [54,55], but equivalently increases the plate length. Therefore, the cylinder-plate body without a gap in this work presents larger amplitudes, which coincides with the results in Stappenbelt [29] and Assi and Bearman [28].

Fig. 7
Fig. 7
Close modal

Figure 8 plots the vibration orbits of the bare cylinder and cylinder with a rear splitter plate. The figure-eight and C-shape orbits demonstrate the in-line vibration frequencies are double that of the cross-flow ones. For the bare cylinder, the evolution of Lissajous diagrams from C-shape to figure-eight corresponds to the phase difference shifts from 0 to π. In contrast, the orbits evolve back into a C-shape in the galloping regime when a rear splitter plate is equipped, in accompany by the phase difference drops from π to 0 [4].

Fig. 8
Fig. 8
Close modal

Usually, the vibration in cross-flow direction with larger amplitudes is considered as the main reason for structural fatigue, compared to the in-line. Therefore, the evaluation of the cross-flow vibration is more meaningful. The time series of fluctuating displacements is analyzed by fast Fourier transform algorithm. In order to clarify the vibration response modes, the peak frequency of the spectrum is normalized by natural frequency, i.e., fY* = fY/fn, where fY is the vibration frequency in cross-flow direction. Figure 9 compares the cross-flow response frequency of a cylinder with or without a rear splitter plate. For the bare cylinder, the response frequency basically follows the Strouhal number of a stationary bare cylinder (St0 = 0.153) in IB, and then locks onto the natural frequency (fY* = 1) in LB, finally follows the St-law again when it enters DB. By introducing a rear splitter plate, the frequency response becomes distinctly different from that of a bare cylinder. First, the lock-in region (LB) is narrowed, indicating the VIV suppression. Second, the frequency drops abruptly to be lower than the natural, further confirming the galloping regime. The onset reduced velocities are Ur = 9, Ur = 11, Ur = 11, and Ur = 12 for 0.5D, 1D, 1.5D, and 2D plates, respectively, which are consistent with the performances of lift coefficient (Fig. 4) as well as amplitude (Fig. 7). Moreover, the longer the plate, the lower the frequency in galloping region, which aligns with the report by Sahu et al. [19]. Although the VIV region becomes narrow after placing a splitter plate behind the cylinder, it still possesses complete branches, including IB, LB, and DB. The galloping region is closely followed the DB, demonstrating a weak coupling response between VIV and galloping [25,26].

Fig. 9
Fig. 9
Close modal

Figure 10 plots the variation of added mass coefficient (Ca) against the reduced velocity for a cylinder with and without a rear splitter plate. It is seen that the added mass coefficient decreases sharply as Ur increases in the IB, then shifts to a negative value in the LB, and continuously decreases in the DB. In contrast, Ca returns back to a positive value in the galloping region, which agrees well with the reported results in Xu et al. [56] and Song et al. [57].

Fig. 10
Fig. 10
Close modal

#### 4.1.3 Wake-Flow Structure.

To understand the galloping mechanism, Fig. 11 presents the recirculation region around the structure and the distribution of pressure coefficient along the plate surface. The recirculation region is enveloped by u = 0, i.e., u < 0 within recirculation region (partially or wholly covering the splitter plate) and u > 0 outside recirculation region. The instantaneous moment is when the body returns to the equilibrium location during the upward motion.

Fig. 11
Fig. 11
Close modal

At Ur = 9, the cylinder with a 0.5D plate experiences large movement, and the galloping mechanism has developed, while other structures remain small amplitudes in desynchronization branch. It is clearly seen that the recirculation region in galloping regime is not symmetric about the centerline of y = 0. The upper shear layer is drawn closer to the plate and reattaches to the plate’s upper surface, introducing an asymmetric pressure distribution along the two sides. The huge pressure difference is considered as the main reason of that the structure would be able to extract more energy from the ambient flow [28,30], and hence strengthen the vibration response as galloping appears. On the contrary, the recirculation region is approximately symmetric about y = 0 for cylinders with 1D, 1.5D, and 2D plates, indicating the vortex formation and shedding are pushed to a further downstream position. Additionally, the pressure distribution along the plate’s upper and lower surfaces are much close, resulting in a near-zero force in the transverse direction.

At large reduced velocities, the cylinder-plate body is easier to gallop due to the reattachment of shear layers, leading to the associated asymmetric pressure distribution. As Ur increases, the reattachment point moves forward, indicating the vortex shedding occurs very earlier and thereby the enhanced vibration. However, the reattachment point moves downward with increasing plate length, accompanied by the pressure decrease between the plate’s two sides, bringing about relatively small vibration amplitude as shown in Fig. 7.

The time-averaged flow fields are presented to evaluate the vortex formation. The iso-contours of root-mean-squared streamwise velocities (u*rms = urms/uin) and contour line of umean = 0 are compared in Fig. 12. The streamwise length (Lr* = Lr/D) of the recirculation region (bubble) is clearly identified by umean = 0. The streamwise distance between cylinder center and the rolling position of shear layers is defined as vortex formation length Lv* (=Lv/D), and the transverse distance between two peaks of u*rms is defined as wake width W* = W/D. At Ur = 6, Lr* and Lv* are distinctly increased by installing a rear splitter plate, and the longer plate leads to more growth in lengths. The wake width is reduced in the presence of a splitter plate. These variations result in the reduction of hydrodynamic forces and hence the VIV suppression. However, the wake width of the cylinder-plate is significantly increased at Ur = 18. The vortex formation length increases with increasing plate length, and finally larger than that of the bare cylinder, indicating the longer plate could push vortex shedding to a further downstream position. Both the shorter recirculation region length of cylinder-plate body and reattachment behavior suggest the galloping phenomenon.

Fig. 12
Fig. 12
Close modal

### 4.2 Effect of Plate Location.

In this section, two more splitter plates of Lu/D = 0.5 and 1 are arranged in front of the cylinder to study the effect of plate location. Figure 13 compares the hydrodynamic coefficients of a cylinder with splitter plates placed upstream and downstream individually and simultaneously. Although the time-averaged drag coefficient is reduced when the cylinder is attached to a front plate, the rise and fall behavior are still observed, just like the trend of the bare cylinder. Moreover, the drag coefficient can be further reduced when considering the bilateral arrangement in the whole reduced velocity range, and the longer the rear plate, the larger the drag-reduction. It is noted that there is a distinct peak region (Ur = 4–9) of drag coefficient for the cylinder with 0.5D rear plate compared with other cases, which is related to the hydrodynamic instability and will be discussed later. When it comes to CL,rms, the cylinder with an individual front plate follows the trend of bare cylinder, despite the decrease in value. Although there are also rise and fall of CL,rms for the cylinder with 0.5D rear plate, it is totally different from a typical VIV response, which includes the initial, lower, and desynchronization branches. Instead, the lower branch is replaced by a galloping branch, and this interesting finding will be explained from the perspectives of response frequency and the phase difference between cross-flow displacement and lift coefficients later. For the cylinder with 0.5D front plate and 1D, 1.5D rear plate, sharp rises are observed at Ur = 10–11 and Ur = 14–15, respectively, and then the lift coefficient remains a quite larger value as compared to bare cylinder. It confirms the occurrence of hydrodynamic instability and thus galloping response. Further increasing rear plate to 2D, the abrupt rise of CL,rms disappears, suggesting the absence of galloping. When considering the cylinder with 1D front plate and 1D, 1.5D, and 2D rear plate, obvious global lift-reduction can be observed, which is helpful to suppress the vibration response.

Fig. 13
Fig. 13
Close modal

Figure 14 compares the response amplitude of the cylinder with splitter plates placed upstream and downstream individually and simultaneously. Considering the group of 0.5D front plate, three vibration patterns are observed: (1) typical VIV response including IB, LB, and DB for bare cylinder, cylinder with an individual front 0.5D plate, and cylinder with 0.5D front plate and 2D rear plate; (2) IB-galloping-DB response for cylinder with 0.5D front plate and 0.5D rear plate, where the galloping branch is observed at Ur = 5–11; and (3) weak VIV-galloping coupling response, i.e., IB-LB-DB-galloping response for cylinder with 0.5D front plate and 1D, 1.5D rear plate. For the group of 1D front plate, the IB-galloping-DB response still exists in the case of 0.5D rear plate, but the galloping region is distinctly shrunk (Ur = 5–9). Moreover, the low interference between VIV and galloping observed at the case of cylinder with 0.5D front plate and 1D, 1.5D rear plate has been replaced by the typical VIV response. By introducing a longer front plate of Lu/D = 1, a better vibration suppression is achieved, which coincides with our earlier results [17], further confirming that a streamlined profile is helpful to suppress the structure’s vibration.

Fig. 14
Fig. 14
Close modal

To explain the aforementioned vibration responses, Figs. 15 and 16 depict the vibration frequency spectra in cross-flow direction for the groups of cylinder with a 0.5D and 1D front plate, respectively. As Fig. 15 shows, the variations of response frequency for a cylinder with an individual 0.5D front plate basically follow the trend of St = 0.153 at Ur = 3–6, then lock on at Ur = 7, and follow St-law at Ur = 8–18 again, indicating a typical VIV response. When a 0.5D plate is arranged behind the cylinder simultaneously, the lock-in region is disappeared, while the galloping response, characterized by a lower-frequency than the natural frequency, is observed at Ur = 5–11. Further increasing the rear plate to 1D or 1.5D, the galloping region followed by DB occurs at Ur = 11 and 15, respectively, and the complete VIV region is observed at the same time. The typical VIV response frequency pattern occurs once again when the cylinder is equipped with a 0.5D front plate and a 2D rear plate simultaneously.

Fig. 15
Fig. 15
Close modal
Fig. 16
Fig. 16
Close modal

Now we use the phase indifference (φ) between cross-flow displacement and lift force to further explain the appearance of different vibration patterns found in Fig. 14. Three typical vibration patterns are compared in Fig. 17, including the typical VIV pattern, IB-galloping-DB pattern, and weak VIV-galloping coupling response pattern, which are exemplified by the cases of Lu/D = 0.5 + Ld/D = 2, Lu/D = 0.5 + Ld/D = 0.5, and Lu/D = 0.5 + Ld/D = 1, respectively. As Assi and Bearman [28] stated, the resonant response of VIV presents a phase shift of almost 180 deg when the response crosses the resonance peak (between Ur = 5 and 9). However, the galloping response experiences no such shift. Instead, the phase difference remains at low values for the whole galloping response range with lift leading the movement of the cylinder by a very small time lag. This small value of phase difference, coupled with the lower-frequency than natural frequency (as shown in Figs. 15 and 16), proves the appearance of high-amplitude galloping vibrations. Moreover, the reattachment phenomenon is distinctly observed at the upper side of rear plate when galloping occurs, introducing a relatively huge pressure difference between the plate’s two surfaces and thus a large lift force, as shown in Fig. 17(b). By comparison, the pressure distributions along two surfaces of the rear plate are pretty close when the cylinder enters the DB region, yielding a near-zero force in the transverse direction and therefore a very small amplitude.

Fig. 17
Fig. 17
Close modal

## 5 Conclusions

In this work, two-dimensional numerical simulations were conducted to investigate the flow-induced vibration response of a circular cylinder attached with front or/and rear splitter plates at a low Reynolds number of 120. The major conclusions are drawn as follows:

1. With the introduction of a rear splitter plate, both the time-averaged drag coefficient and RMS lift coefficient are significantly reduced at Ur ≤ 8, contributing to the VIV suppression. Nevertheless, the galloping is excited at Ur > 8 as a result of the occurrence of hydrodynamic instability. Such a response pattern is termed a weak VIV-galloping coupling response. The onset of galloping is associated with the rapid jump of response amplitude and hydrodynamic force as well as the abrupt drop in response frequency. The longer the splitter plate, the higher the onset reduced velocity of galloping. In the galloping region, the rear recirculation bubble and the vortex formation length are shortened as compared to that in the VIV region, accompanied by the alternate reattachment of shear layers on the plate surface. Therefore, the cylinder-plate body introduces an extra lift force that strengthens the vibration response.

2. With the attachment of an individual front plate, the reduction of hydrodynamic force is achieved at the considered reduced velocity range and thereby the VIV suppression. The hydrodynamic coefficient and response amplitude follow the trend of bare cylinder, presenting the typical initial, lower, and desynchronization branches, but the lock-in region is narrowed.

3. For a cylinder with splitter plates attached upstream and downstream simultaneously, three typical vibration patterns are observed: VIV, weak VIV-galloping coupling response, and IB-galloping-DB mode. The galloping branch in IB-galloping-DB mode occurs with an abrupt drop in response frequency as well as a tiny time lag between the displacement and the lift force. Additionally, the alternate reattachment of boundary layers is distinctly observed at one side of the rear plate, resulting in a relatively large pressure difference between the two surfaces and hence a large lift force. Nevertheless, the best VIV suppression is achieved by the bilateral plates of Lu/D = 1 and Ld/D = 1, 1.5, and 2 due to the more streamlined profiles.

Further extensive investigations are required to systematically examine the effects of more related parameters on the FIV of cylinder-plate body, such as length ratio of splitter plate, Re, and mass-damping ratio.

## Acknowledgment

This research was supported by the National Natural Science Foundation of China (Grant No. 51979238). The work was carried out in the computer cluster of the laboratory of offshore oil and gas engineering at Southwest Petroleum University (SWPU).

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

## References

1.
Wang
,
J. L.
,
Zhou
,
S. X.
,
Zhang
,
Z. E.
, and
Yurchenko
,
D.
,
2019
, “
High-Performance Piezoelectric Wind Energy Harvester With Y-Shaped Attachments
,”
Energy Convers. Manage.
,
181
, pp.
645
652
.
2.
Wang
,
J. L.
,
Geng
,
L. F.
,
Zhou
,
S. X.
,
Zhang
,
Z. E.
,
Lai
,
Z. H.
, and
Yurchenko
,
D.
,
2020
, “
Design, Modeling and Experiments of Broadband Tristable Galloping Piezoelectric Energy Harvester
,”
Acta Mech. Sinica.
,
36
(
3
), pp.
592
605
.
3.
Yang
,
K.
,
Wang
,
J. L.
, and
Yurchenko
,
D.
,
2019
, “
A Double-Beam Piezo-Magneto-Elastic Wind Energy Harvester for Improving the Galloping-Based Energy Harvesting
,”
Appl. Phys. Lett.
,
115
(
19
), p.
193901
.
4.
Zhu
,
H. J.
,
Tang
,
T.
,
Gao
,
Y.
,
Zhou
,
T. M.
, and
Wang
,
J. L.
,
2021
, “
Flow-Induced Vibration of a Trapezoidal Cylinder Placed at Typical Flow Orientations
,”
J. Fluids Struct.
,
103
, p.
103291
.
5.
Zhu
,
H. J.
,
Tang
,
T.
,
Zhou
,
T. M.
,
Liu
,
H. Y.
, and
Zhong
,
J. W.
,
2020a
, “
Flow Structures Around Trapezoidal Cylinders and Their Hydrodynamic Characteristics: Effects of the Base Length Ratio and Attack Angle
,”
Phys. Fluids
,
32
(
10
), p.
103606
.
6.
Choi
,
H.
,
Jeon
,
W. P.
, and
Kim
,
J.
,
2008
, “
Control of Flow Over a Bluff Body
,”
Annu. Rev. Fluids Mech.
,
40
(
1
), pp.
113
139
.
7.
Owen
,
J. C.
,
Bearman
,
P. W.
, and
Szewczyk
,
A. A.
,
2001
, “
Passive Control of VIV With Drag Reduction
,”
J. Fluids Struct.
,
15
(
3–4
), pp.
597
605
.
8.
Rashidi
,
S.
,
Hayatdavoodi
,
M.
, and
Esfahani
,
J. A.
,
2016
, “
Vortex Shedding Suppression and Wake Control: A Review
,”
Ocean Eng.
,
126
, pp.
57
80
.
9.
Zdravkovich
,
M. M.
,
1981
, “
Review and Classification of Various Aerodynamic and Hydrodynamic Means for Suppressing Vortex Shedding
,”
J. Wind Eng. Ind. Aerod.
,
7
(
2
), pp.
145
189
.
10.
Roshko
,
A.
,
1954
, “
On the Drag and Shedding Frequency of Two-Dimensional Bluff Bodies
,”
NACA Technical Notes
.
11.
Apelt
,
C. J.
,
West
,
G. S.
, and
Szewczyk
,
A. A.
,
1973
, “
The Effects of Wake Splitter Plates on the Flow Past a Circular Cylinder in the Range l04 < R < 5 × l04
,”
J. Fluids Mech.
,
61
(
1
), pp.
187
198
.
12.
Fiedler
,
H. E.
, and
Fernholz
,
H. H.
,
1990
, “
On Management and Control of Turbulent Shear Flows
,”
Prog. Aerosp. Sci.
,
27
(
4
), pp.
305
387
.
13.
Kawai
,
H.
,
1990
, “
Discrete Vortex Simulation for Flow Around a Circular Cylinder With a Splitter Plate
,”
J. Wind Eng. Ind. Aerodyn.
,
33
(
1–2
), pp.
153
160
.
14.
Bearman
,
P. W.
,
1965
, “
Investigation of the Flow Behind a Two-Dimensional Model With a Blunt Trailing Edge and Fitted With Splitter Plates
,”
J. Fluids Mech.
,
21
(
2
), pp.
241
255
.
15.
Kwon
,
K.
, and
Choi
,
H.
,
1996
, “
Control of Laminar Vortex Shedding Behind a Circular Cylinder Using Splitter Plate
,”
Phys. Fluids
,
8
(
2
), p.
479
.
16.
Schaefer
,
J. W.
, and
Eskinazi
,
S.
,
1959
, “
An Analysis of the Vortex Street Generated in a Viscous Fluid
,”
J. Fluids Mech.
,
6
(
2
), pp.
241
260
.
17.
Zhu
,
H. J.
,
Li
,
G. M.
, and
Wang
,
J. L.
,
2020b
, “
Flow-Induced Vibration of a Circular Cylinder With Splitter Plates Placed Upstream and Downstream Individually and Simultaneously
,”
Appl. Ocean Res.
,
97
, p.
102084
.
18.
Hu
,
Z. M.
,
Wang
,
J. S.
,
Sun
,
Y. K.
, and
Zheng
,
H. X.
,
2021
, “
Flow-Induced Vibration Suppression for a Single Cylinder and One-Fixed-One-Free Tandem Cylinders With Double Tail Splitter Plates
,”
J. Fluids Struct.
,
106
, p.
103373
.
19.
Sahu
,
T. R.
,
Furquan
,
M.
,
Jaiswal
,
Y.
, and
Mittal
,
S.
,
2019
, “
Flow-Induced Vibration of a Circular Cylinder With Rigid Splitter Plate
,”
J. Fluids Struct.
,
89
, pp.
244
256
.
20.
Parkinson
,
G.
,
1989
, “
Phenomena and Modelling of Flow-Induced Vibrations of Bluff Bodies
,”
Prog. Aerosp. Sci
,
26
(
2
), pp.
169
224
.
21.
Sarpkaya
,
T.
,
1979
, “
Vortex-Induced Oscillations
,”
ASME J. Appl. Mech.
,
46
(
2
), pp.
241
258
.
22.
Bearman
,
P. W.
,
1984
, “
Vortex Shedding From Oscillating Bluff Bodies
,”
Annu. Rev. Fluids Mech.
,
16
(
1
), pp.
195
222
.
23.
Nakamura
,
Y.
,
1988
, “
Recent Research Into Bluff-Body Flutter
,”
J. Wind Eng. Ind. Aerodyn.
,
33
(
1–2
), pp.
1
10
.
24.
Parkinson
,
G. V.
,
1989
, “
Phenomenon and Modelling of Flow-Induced Vibrations of Bluff Bodies
,”
Prog. Aerosp. Sci.
,
26
(
2
), pp.
169
224
.
25.
Mannini
,
C.
,
Marra
,
A. M.
, and
Bartoli
,
G.
,
2014
, “
VIV-Galloping Instability of Rectangular Cylinders: Review and New Experiments
,”
J. Wind Eng. Ind. Aerodyn.
,
132
, pp.
109
124
.
26.
Mannini
,
C.
,
Marra
,
A. M.
,
Massai
,
T.
, and
Bartoli
,
G.
,
2016
, “
Interference of Vortex-Induced Vibration and Transverse Galloping for a Rectangular Cylinder
,”
J. Fluids Struct.
,
66
, pp.
403
423
.
27.
Assi
,
G. R. S.
,
Bearman
,
P. W.
, and
Kitney
,
N.
,
2009
, “
Low Drag Solutions for Suppressing Vortex-Induced Vibration of Circular Cylinder
,”
J. Fluids Struct.
,
25
(
4
), pp.
666
675
.
28.
Assi
,
G. R. S.
, and
Bearman
,
P. W.
,
2015
, “
Transverse Galloping of Circular Cylinders Fitted With Solid and Slotted Splitter Plates
,”
J. Fluids Struct.
,
54
, pp.
263
280
.
29.
Stappenbelt
,
B.
,
2010
, “
Splitter-Plate Wake Stabilisation and Low Aspect Ratio Cylinder Flow-Induced Vibration Mitigation
,”
Int. J. Offshore Polar
,
20
(
3
), pp.
1
6
.
30.
Liang
,
S. P.
,
Wang
,
J. S.
, and
Hu
,
Z. M.
,
2018
, “
VIV and Galloping Response of a Circular Cylinder With Rigid Detached Splitter Plates
,”
Ocean Eng.
,
162
, pp.
176
186
.
31.
Gao
,
D. L.
,
Chen
,
G. B.
,
Huang
,
Y. W.
,
Chen
,
W. L.
, and
Li
,
H.
,
2020
, “
Flow Characteristics of a Fixed Circular Cylinder With an Upstream Splitter Plate: On the Plate-Length Sensitivity
,”
Exp. Therm. Fluids Sci.
,
117
, p.
110135
.
32.
Chutkey
,
K.
,
Suriyanarayanan
,
P.
, and
Venkatakrishnan
,
L.
,
2018
, “
Near Wake Field of Circular Cylinder With a Forward Splitter Plate
,”
J. Wind Eng. Ind. Aerod.
,
173
, pp.
28
38
.
33.
Qiu
,
Y.
,
Sun
,
Y.
,
Wu
,
Y.
, and
Tamura
,
Y.
,
2014
, “
Effects of Splitter Plates and Reynolds Number on the Aerodynamic Loads Acting on a Circular Cylinder
,”
J. Wind Eng. Ind. Aerod.
,
127
, pp.
40
50
.
34.
,
K.
,
Vandiver
,
J. K.
, and
Larsen
,
C. M.
,
2000
, “
Added Mass and Oscillation Frequency for a Circular Cylinder Subjected to Vortex-Induced Vibrations and External Disturbance
,”
J. Fluids Struct.
,
14
(
7
), pp.
1071
1088
.
35.
Jiang
,
H. Y.
,
Cheng
,
L.
,
Draper
,
S.
,
An
,
H. W.
, and
Tong
,
F. F.
,
2016
, “
Three-Dimensional Direct Numerical Simulation of Wake Transitions of a Circular Cylinder
,”
J. Fluids Mech.
,
801
, pp.
353
391
.
36.
Zhu
,
H. J.
,
Tang
,
T.
,
Zhao
,
H. L.
, and
Gao
,
Y.
,
2019
, “
Control of Vortex-Induced Vibration of a Circular Cylinder Using a Pair of Air Jets at Low Reynolds Number
,”
Phys. Fluids
,
31
(
4
), p.
043603
.
37.
Zhu
,
H. J.
,
Yao
,
J.
,
Ma
,
Y.
,
Zhao
,
H. N.
, and
Tang
,
Y. B.
,
2015
, “
Simultaneous CFD Evaluation of VIV Suppression Using Smaller Control Cylinders
,”
J. Fluids Struct.
,
57
, pp.
66
80
.
38.
Zhu
,
H. J.
,
Zhao
,
H. N.
,
Yao
,
J.
, and
Tang
,
Y. B.
,
2016
, “
Numerical Study on Vortex-Induced Vibration Responses of a Circular Cylinder Attached by a Free-to-Rotate Dartlike Overlay
,”
Ocean Eng.
,
112
, pp.
195
210
.
39.
Zhu
,
H. J.
, and
Gao
,
Y.
,
2018
, “
Hydrokinetic Energy Harvesting From Flow-Induced Vibration of a Circular Cylinder With Two Symmetrical Fin-Shaped Strips
,”
Energy.
,
165
, pp.
1259
1281
.
40.
Zhu
,
H. J.
,
Zhao
,
Y.
, and
Zhou
,
T. M.
,
2018
, “
Numerical Investigation of the Vortex-Induced Vibration of an Elliptic Cylinder Free-to-Rotate About Its Center
,”
J. Fluids Struct.
,
83
, pp.
133
155
.
41.
Bao
,
Y.
,
Zhou
,
D.
, and
Tu
,
J. H.
,
2011
, “
Flow Interference Between a Stationary Cylinder and an Elastically Mounted Cylinder Arranged in Proximity
,”
J. Fluids Struct.
,
27
(
8
), pp.
1425
1446
.
42.
Wu
,
J.
,
Shu
,
C.
, and
Zhao
,
N.
,
2014
, “
Numerical Investigation of Vortex-Induced Vibration of a Circular Cylinder With a Hinged Flat Plate
,”
Phys. Fluids
,
26
(
6
), p.
063601
.
43.
Borazjani
,
I.
, and
Sotiropoulos
,
F.
,
2009
, “
Vortex-Induced Vibrations of Two Cylinders in Tandem Arrangement in the Proximity-Wake Interference Region
,”
J. Fluids Mech.
,
621
, pp.
321
364
.
44.
Williamson
,
C. H. K.
, and
Govardhan
,
R.
,
2004
, “
Vortex-Induced Vibrations
,”
Annu. Rev. Fluids Mech.
,
36
(
1
), pp.
413
455
.
45.
Jauvtis
,
N.
, and
Williamson
,
C. H. K.
,
2003
, “
Vortex-Induced Vibration of a Cylinder With Two Degrees of Freedom
,”
J. Fluids Struct.
,
17
(
7
), pp.
1035
1042
.
46.
Sudhakar
,
Y.
, and
,
S.
,
2012
, “
Vortex Shedding Characteristics of a Circular Cylinder With an Oscillating Wake Splitter Plate
,”
Comput. Fluids
,
53
, pp.
40
52
.
47.
Lu
,
L.
,
Guo
,
X. L.
,
Tang
,
G. Q.
,
Liu
,
M. M.
,
Chen
,
C. Q.
, and
Xie
,
Z. H.
,
2016
, “
Numerical Investigation of Flow-Induced Rotary Oscillation of Circular Cylinder With Rigid Splitter Plate
,”
Phys. Fluids
,
28
(
9
), p.
093604
.
48.
Hwang
,
J. Y.
,
Yang
,
K. S.
, and
Sun
,
S. H.
,
2003
, “
Reduction of Flow-Induced Forces on a Circular Cylinder Using a Detached Splitter Plate
,”
Phys. Fluids
,
15
(
8
), pp.
2433
2436
.
49.
Deep
,
D.
,
Sahasranaman
,
A.
, and
Senthilkumar
,
S.
,
2022
, “
POD Analysis of the Wake Behind a Circular Cylinder With Splitter Plate
,”
Eur. J. Mech. B/Fluids
,
93
, pp.
1
12
.
50.
Achenbach
,
E.
,
1971
, “
Influence of Surface Roughness on the Cross-Flow Around a Circular Cylinder
,”
J. Fluids Mech.
,
46
(
2
), pp.
321
335
.
51.
Akilli
,
H.
,
Karakus
,
C.
,
Akar
,
A.
,
Sahin
,
B.
, and
Tumen
,
N. F.
,
2008
, “
Control of Vortex Shedding of Circular Cylinder in Shallow Water Flow Using an Attached Splitter Plate
,”
ASME J. Fluids Eng.
,
130
(
4
), p.
041401
.
52.
Zhao
,
M.
,
Cheng
,
L.
, and
Zhou
,
T. M.
,
2013
, “
Numerical Simulation of Vortex-Induced Vibration of a Square Cylinder at a Low Reynolds Number
,”
Phys. Fluids
,
25
(
2
), p.
023603
.
53.
Jaiman
,
R. K.
,
Guan
,
M. Z.
, and
Miyanawala
,
T. P.
,
2016
, “
Partitioned Iterative and Dynamic Subgrid-Scale Methods for Freely Vibrating Square-Section Structures at Subcritical Reynolds Number
,”
Comput. Fluids
,
133
, pp.
68
89
.
54.
Ozono
,
S.
,
1999
, “
Flow Control of Vortex Shedding by a Short Splitter Plate Asymmetrically Arranged Downstream of a Cylinder
,”
Phys. Fluids
,
11
(
10
), pp.
2928
2934
.
55.
Mittal
,
S.
,
2003
, “
Effect of ‘Slip’ Splitter Plate on Vortex Shedding From a Cylinder
,”
Phys. Fluids
,
15
(
3
), pp.
817
820
.
56.
Xu
,
W. H.
,
Ji
,
C. N.
,
Sun
,
H.
,
Ding
,
W. J.
, and
Bernitsas
,
M. M.
,
2019
, “
Flow-Induced Vibration of Two Elastically Mounted Tandem Cylinders in Cross-Flow at Subcritical Reynolds Numbers
,”
Ocean Eng.
,
173
, pp.
375
387
.
57.
Song
,
L. J.
,
Fu
,
S. X.
,
Cao
,
J.
,
Ma
,
L. X.
, and
Wu
,
J. Q.
,
2016
, “
An Investigation Into the Hydrodynamics of a Flexible Riser Undergoing Vortex-Induced Vibration
,”
J. Fluids Struct.
,
63
, pp.
325
350
.