## 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 U_{r} ≤ 8 (U_{r} is the reduced velocity), resulting in the vortex-induced vibration (VIV) suppression. Nevertheless, the galloping is excited at U_{r} > 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 [1–5]. 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 [10–12]. 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 l0^{4} < Re < 5 × l0^{4}, 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 [13,17–19]. Vortex-induced vibration (VIV) and galloping are the most common patterns of FIV. Various papers have discussed these two vibrations [20–23], as well as their interaction [24–26]. 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 0.25–2.0*D* 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 0.5*D*. 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 1*D* 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 (Re_{c}) of 3.0 × 10^{5}–3.31 × 10^{5} to achieve drag-reduction. The arrangements of a front plate or bilateral plates perform better at Re < Re_{c}. 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

*D*. The front and rear splitter plate lengths are defined as

*L*

_{u}and

*L*

_{d}, respectively. The plate thickness is set as

*δ*= 0.1

*D*. 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*/

*m*, and

_{d}*m*,

*m*represent the structure mass and the displaced mass of fluid, respectively. The damping ratio

_{d}*ζ*is set as 0.01, where $\zeta =C/(2K(m+ma))$, and

*m*,

_{a}*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

*U*

_{r}= 3–18, the natural frequency

*f*

_{n}of the present cylinder-plate body in still water is set as a variable by varying the system damping accordingly, expressed as

*m*=

_{a}*C*,

_{a}m_{d}*C*is the added mass coefficient and one can see more detailed information and calculating method about

_{a}*C*in the report by Vikestad et al. [34]. As a result, the variation of

_{a}*U*

_{r}is achieved without the change of Re. The main simulation parameters in this work are listed in Table 1.

Parameter | Symbol | Value |
---|---|---|

Mass ratio | m* | 6.9 |

Damping ratio | ζ | 0.01 |

Length of the front splitter plate | L_{u}/D | 0.5, 1 |

Length of the rear splitter plate | L_{d}/D | 0.5, 1, 1.5, 2 |

Thickness of the splitter plate | δ/D | 0.1 |

Reduced velocity | U_{r} | 3–18 |

Reynolds number | Re | 120 |

Parameter | Symbol | Value |
---|---|---|

Mass ratio | m* | 6.9 |

Damping ratio | ζ | 0.01 |

Length of the front splitter plate | L_{u}/D | 0.5, 1 |

Length of the rear splitter plate | L_{d}/D | 0.5, 1, 1.5, 2 |

Thickness of the splitter plate | δ/D | 0.1 |

Reduced velocity | U_{r} | 3–18 |

Reynolds number | Re | 120 |

As displayed in Fig. 1(a), the computational domain is a rectangular region with a size of 70*D* (in-line direction) × 40*D* (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 20*D* from the upstream and two lateral boundaries. For the upstream boundary, a uniform velocity profile is specified as *u* = *u*_{in} 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 6*D* concentric with the circular cylinder, combined with a rectangle region of 6*D* × 4*D*, is defined as an accompanying moving zone, which follows the structure’s movement. A slightly larger rectangle region (25*D* × 20*D*) 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.

*x*is the Cartesian coordinate in

_{i}*i*direction,

*u*is the velocity component in the direction

_{i}*x*,

_{i}*u*

_{j}^{m}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].

*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,

*f*

_{D}(

*t*) and

*f*

_{L}(

*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]:

*S*

_{1},

*S*

_{2},

*S*

_{3}, and

*S*

_{4}are the intermediate functions, Δ

*t*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.

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 (*A*_{X}* = *A*_{X}/*D* and *A*_{Y}* = *A*_{Y}/*D*, where *A*_{X} and *A*_{Y} 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 0.0004*D*, corresponding to the requirement of *y*^{+} < 1.

Mesh | Nodes on the surface of the cylinder-plate body | Elements | A_{Y}* | A_{X}* |
---|---|---|---|---|

M1 | 300 | 50,582 | 0.82679 (8.61%) | 0.03235 (6.29%) |

M2 | 346 | 81,730 | 0.8826 (2.45%) | 0.03397 (1.59%) |

M3 | 392 | 113,328 | 0.90122 (0.39%) | 0.03443 (0.26%) |

M4 | 434 | 145,808 | 0.90473 | 0.03452 |

Mesh | Nodes on the surface of the cylinder-plate body | Elements | A_{Y}* | A_{X}* |
---|---|---|---|---|

M1 | 300 | 50,582 | 0.82679 (8.61%) | 0.03235 (6.29%) |

M2 | 346 | 81,730 | 0.8826 (2.45%) | 0.03397 (1.59%) |

M3 | 392 | 113,328 | 0.90122 (0.39%) | 0.03443 (0.26%) |

M4 | 434 | 145,808 | 0.90473 | 0.03452 |

### 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 1*D* length is attached to the rear base of the cylinder. The validations are made for the time-averaged coefficient (*C*_{D,mean}), Strouhal number (St), and the normalized length of the recirculation bubble (*L*_{r}* = *L*_{r}/*D*, where *L*_{r} 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, *C*_{D,mean}, and *L*_{r}* 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.

## 4 Results and Discussion

### 4.1 Effect of Plate Length

#### 4.1.1 Hydrodynamic Coefficient.

*L*

_{d}/

*D*= 0.5, 1, 1.5, and 2 against the bare cylinder are plotted in Fig. 4. The time-averaged drag coefficient (

*C*

_{D,mean}) and the root-mean-squared lift coefficient (

*C*

_{L,rms}) are calculated by

*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

*C*

_{D,mean}and

*C*

_{L,rms}of the bare cylinder increase first as

*U*

_{r}increases from 3, and they peak at

*U*

_{r}= 6 and 5, respectively, followed by a drop and then remain a relatively stable value at 8 ≤

*U*

_{r}≤ 18. From the

*C*

_{D,mean}–

*U*

_{r}and

*C*

_{L,rms}–

*U*

_{r}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 1

*D*, 1.5

*D*, and 2

*D*, the drag forces are completely suppressed in the whole

*U*

_{r}range in comparison with the bare cylinder. However, in this work, a drag-increase phenomenon is observed for a shorter plate of

*L*

_{d}/

*D*= 0.5 at 9 ≤

*U*

_{r}≤ 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 0.5

*D*. Nevertheless, the drag forces of the cylinder-plate body decrease as plate length increases from 0.5

*D*, which is due mainly to the improved streamlining of the structure.

Compared to the bare cylinder, the root-mean-squared lift coefficient is obviously diminished at *U*_{r} ≤ 8, 10, 10, and 11, when the cylinder is equipped with a splitter plate of *L*_{d}/*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 *U*_{r} = 5 for all considered cases, presenting a typical VIV response [28]. After a relatively stable stage, *C*_{L,rms} of the cylinder-plate body abruptly rise around *U*_{r} = 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 *C*_{L,rms} is attributed to the hydrodynamic instability which usually causes the galloping response. Besides, the onsets of hydrodynamic instability and variations of *C*_{L,rms} at galloping range are closely related to plate length. For a 0.5*D* rear plate, the rapid rise of *C*_{L,rms} occurs at *U*_{r} = 8–9, after that a monotonously linear decline is observed. The onset of hydrodynamic instability emerges at *U*_{r} = 10–11 both for 1*D* and 1.5*D* rear plate. However, the deceleration of *C*_{L,rms} for 1.5*D* plate is much slower than that of 1*D* plate. Further increasing the plate length to 2*D*, the abrupt rise of *C*_{L,rms} occurs at a higher *U*_{r} range between 11 and 12. Meanwhile, the variations become relatively stable at *U*_{r} = 13–18, indicating that reduced velocity is less influential. In general, the shorter plates own the smaller onset *U*_{r} of hydrodynamic instability, and the larger decline of *C*_{L,rms} at a high *U*_{r} 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 1*D*. However, the wake flow is more unstable at *L*_{d}/*D* ≤ 1. Therefore, the greater reduction of lift coefficients for the cases of *L*_{d}/*D =* 0.5 and 1 is observed in the galloping regime, as compared to those of *L*_{d}/*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.

*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

*U*

_{r}= 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

*L*

_{d}/

*D*= 0.5 at

*U*

_{r}= 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 0.5

*D*to 2

*D*, the absolute values of negative pressure are decreased with the accompaniment of a shrinking local low-pressure zone. Consequently, the 2

*D*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.

#### 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 *U*_{r} = 3–6, then declines at *U*_{r} = 6–8, and finally keeps relatively stable at *U*_{r} = 8–18, consequently named initial, lower, and desynchronization branches. It is noted that the peak amplitude occurs at *U*_{r} = 6, larger than that of lift coefficient at *U*_{r} = 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 *U*_{r} = 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 0.5*D*, 1*D*, 1.5*D*, and 2*D* 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 *U*_{r} 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].

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].

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., *f*_{Y}* = *f*_{Y}/*f*_{n}, where *f*_{Y} 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 (St_{0} = 0.153) in IB, and then locks onto the natural frequency (*f*_{Y}* = 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 *U*_{r} = 9, *U*_{r} = 11, *U*_{r} = 11, and *U*_{r} = 12 for 0.5*D*, 1*D*, 1.5*D*, and 2*D* 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].

Figure 10 plots the variation of added mass coefficient (*C _{a}*) 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

*U*

_{r}increases in the IB, then shifts to a negative value in the LB, and continuously decreases in the DB. In contrast,

*C*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].

_{a}#### 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.

At *U*_{r} = 9, the cylinder with a 0.5*D* 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 1*D*, 1.5*D*, and 2*D* 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 *U*_{r} 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} = *u*_{rms}/*u*_{in}) and contour line of *u*_{mean} = 0 are compared in Fig. 12. The streamwise length (*L*_{r}* = *L*_{r}/*D*) of the recirculation region (bubble) is clearly identified by *u*_{mean} = 0. The streamwise distance between cylinder center and the rolling position of shear layers is defined as vortex formation length *L*_{v}* (=*L*_{v}/*D*), and the transverse distance between two peaks of *u**_{rms} is defined as wake width *W** = *W*/*D*. At *U*_{r} = 6, *L*_{r}* and *L*_{v}* 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 *U*_{r} = 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.

### 4.2 Effect of Plate Location.

In this section, two more splitter plates of *L*_{u}/*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 (*U*_{r} = 4–9) of drag coefficient for the cylinder with 0.5*D* rear plate compared with other cases, which is related to the hydrodynamic instability and will be discussed later. When it comes to *C*_{L,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 *C*_{L,rms} for the cylinder with 0.5*D* 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 0.5*D* front plate and 1*D*, 1.5*D* rear plate, sharp rises are observed at *U*_{r} = 10–11 and *U*_{r} = 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 2*D*, the abrupt rise of *C*_{L,rms} disappears, suggesting the absence of galloping. When considering the cylinder with 1*D* front plate and 1*D*, 1.5*D*, and 2*D* rear plate, obvious global lift-reduction can be observed, which is helpful to suppress the vibration response.

Figure 14 compares the response amplitude of the cylinder with splitter plates placed upstream and downstream individually and simultaneously. Considering the group of 0.5*D* front plate, three vibration patterns are observed: (1) typical VIV response including IB, LB, and DB for bare cylinder, cylinder with an individual front 0.5*D* plate, and cylinder with 0.5*D* front plate and 2*D* rear plate; (2) IB-galloping-DB response for cylinder with 0.5*D* front plate and 0.5*D* rear plate, where the galloping branch is observed at *U*_{r} = 5–11; and (3) weak VIV-galloping coupling response, i.e., IB-LB-DB-galloping response for cylinder with 0.5*D* front plate and 1*D*, 1.5*D* rear plate. For the group of 1*D* front plate, the IB-galloping-DB response still exists in the case of 0.5*D* rear plate, but the galloping region is distinctly shrunk (*U*_{r} = 5–9). Moreover, the low interference between VIV and galloping observed at the case of cylinder with 0.5*D* front plate and 1*D*, 1.5*D* rear plate has been replaced by the typical VIV response. By introducing a longer front plate of *L*_{u}/*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.

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 0.5*D* and 1*D* front plate, respectively. As Fig. 15 shows, the variations of response frequency for a cylinder with an individual 0.5*D* front plate basically follow the trend of St = 0.153 at *U*_{r} = 3–6, then lock on at *U*_{r} = 7, and follow St-law at *U*_{r} = 8–18 again, indicating a typical VIV response. When a 0.5*D* 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 *U*_{r} = 5–11. Further increasing the rear plate to 1*D* or 1.5*D*, the galloping region followed by DB occurs at *U*_{r} = 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 0.5*D* front plate and a 2*D* rear plate simultaneously.

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 *L*_{u}/*D* = 0.5 + *L*_{d}/*D* = 2, *L*_{u}/*D* = 0.5 + *L*_{d}/*D* = 0.5, and *L*_{u}/*D* = 0.5 + *L*_{d}/*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 *U*_{r} = 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.

## 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:

With the introduction of a rear splitter plate, both the time-averaged drag coefficient and RMS lift coefficient are significantly reduced at

*U*_{r}≤ 8, contributing to the VIV suppression. Nevertheless, the galloping is excited at*U*_{r}> 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.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.

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

*L*_{u}/*D*= 1 and*L*_{d}/*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

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