Control of vortex-induced vibrations of the cylinder by using split-ter plates immersed in the cylinder wake at low Reynolds number ＊

Zhi-qiang Xin, Zhi-huang Wu, Chui-jie Wu, Dan Huang

1. The College of Mechanics and Materials, HoHai University, Nanjing 211100, China

2. State Key Laboratory of Structural Analysis for Industrial Equipment and School of Aeronautics and Astronautics, Dalian University of Technology, Dalian, China

Abstract: The vortex-induced vibrations of a cylinder with two plates symmetrically distributed along the centerline of the wake are studied by using the fluid-structure interaction simulations on the Arbitrary Lagrangian-Eulerian method. In this study, the different geo-metrical distribution parameters of the splitter plates and inflow velocities are taken into account. The physical mechanisms of vortex-induced vibration of the cylinder with symmetrical plates are revealed from the cylinder amplitude, hydrodynamic force characteristics, vortex shedding frequency, and flow pattern modification. The results show that the dynamic interaction between the vortex shed from the oscillation cylinder and plates is responsible for the wake stabilization mechanism, and the stable wake and the dynamic response reduction of the main cylinder can be achieved at a wide range of reduced velocities by placing sym-metrical sheets at suitable locations.

Key words: Cylinder, vortex-induced vibrations, splitter plates, vibration control, flow pattern

Introduction

Vortex-induced vibration (VIV) is a common phenomenon in engineering and in nature. It has been found that the onset of vortex shedding which is responsible for the structural vibration is triggered when the Reynolds number exceeds a critical value.Under specific flow and structure conditions, the existence of vortex-induced vibrations constitutes a widespread problem which is undesirable in many engineering applications. Therefore, developing effective control methods for suppression of vortexinduced vibrations has become an important research topic.

There has been a wide scientific interest in VIV characterization and control[1-3]. Moreover, the flow mechanism of vortex-induced vibration and the distribution of vibration response branches was revealed detailed in the work of Mittal[4-5]. In order to reduce or completely suppress cylinder vibrations,different means have been proposed, which can be generally classified into passive control without power input, active open-loop and active closed-loop control with power input. There are actuators but no sensor in open-loop control, and sensors and actuators in closed-loop control. Active controls were developed in recent years, which were discussed in these articles[6-7]. Due to the passive methods a generally efficient and simple solution is provided for vibration control in real configurations, these methods have traditionally received more attention. These strategies are mainly based on geometry modifications of the object[8-9]and disturbance of the surrounding flow,such as using splitter plates[10]or smaller control cylinders[11], with the aim of changing the vortex shedding process.

The application of sensitivity to localized perturbations in the wake has been widely studied for rigid stationary cylinders and other two-dimensional bodies in recent years. Wake splitter device has been proved to be an efficient passive control method for drag reduction and wake stabilization[2]. The earlier studies on such devices, more precisely splitter plates,were conducted by Roshko[1,12]firstly through experimental research of an attached plate immersed in the cylinder wake. Roshko found that the vertical wake was modified due to the interference of the plate behind the cylinder. The interaction between the shear free layers emanated from both sides of the cylinder was delayed, which leads to an increase in the base pressure and the decrease in the drag force consequently. The significant experimental investigation from Strykowski and Sreenivasan[2]showed that the vortex-shedding behind the cylinder can be effectively suppressed at low Reynolds numbers,through the appropriately positioned small cylinder in the wake, resulting in an offset on the beginning of the instability, towards a higher critical Reynolds number.The numerical simulation experiment was carried out by Kuo et al.[13]In the range of =80-300Re, the passive control of the cylinder wake with two control rods whose diameter wasd=D/8 placed behind the cylinder was studied. They found that the vortex of the cylinder wake still existed after the control rod was placed, but the lift force and drag force were significantly reduced. Moreover, Parezanovi? and Cadot[14]experimented with the wake of a D-shaped cylinder in the turbulent state, showing that apparent variations of shedding frequency can be achieved when the cylinder is placed along with the shear layers,which can modify the interaction between them and vortex formation.

In view of wake control effect of rigid stationary bodies, it can be inferred that sensitivity analyses may provide valuable information for optimal placement of wake perturbing devices in the wake of elastically mounted rigid cylinders. Recently, Jiménez-González and Huera-Huarte[15]had identified the physical mechanisms underlying the VIV response modifications induced by the control cylinders using digital particle image velocimetry (DPIV). They verified that the VIV response with the Reynolds number changing from 9 000 to 260 000 can be reduced by more than 65% by using control cylinders with diameters of only 12% of the main cylinder diameter. The experiments conducted by Akilli et al.[16]revealed that the splitter plates within 1.75Dfrom the edge of the cylinder have a substantial influence on the vortex shedding atRe= 5000, this is similar to the conclusion of Kawai[17]. More recently, the VIV of a cylinder with multiple small control rods atRe=1161.3-6387.1 was studied by Zhu and Yao[18]with numerical simulation. The results had shown that an obvious suppression effect of VIV in a wide range of Reynolds numbers can be achieved by placing the appropriate number of rods at appropriate locations. It can be seen from the above references that there are a number of research results of experiments and numerical simulations for the VIV wake disturbance control at high Reynolds number. However, the first locking interval in which VIV occurs can also be in the range of lower Reynolds number. Therefore, how to efficiently suppress the wake intensity and cylinder amplitude by the wake disturbance device is still an open issue for the researchers.

In this paper, we numerically investigate the system response modification induced by the downstream wake disturbance, which is exerted by two splitter plates fixed symmetrically along the wake centerline. The effect of a wide range of geometry parameters on the VIV of a cylinder, include the gap distance between cylinder and plates as well as the vertical distance between plates, and reduced velocity are analyzed. It is anticipated that this investigation could provide valuable insight into wake control in the VIV at low Reynolds number and improve our understanding of the complicated wake-body interaction problem.

1. Problem definition

The control of VIV of the cylinder with two splitter plates in the cylinder wake fields is studied in this paper. The schematic configuration of the flow model was shown in Fig. 1. A circular cylinder of diameterDwith two splitter plates was immersed in a uniform flow with a streamwise velocity ofU.Two rectangle plates with side length ofL(equal to the cylinder diameter,L=D) and thickness ratio of 0.1 was placed symmetrically along the center line at a vertical distanceHand downstream distanceSfrom the cylinder.

Fig. 1 (Color online) Geometry configuration of the flow over an elastically mounted cylinder with two stationary splitter plates

The cylinder was simplified as a single degree of freedom spring oscillator model. The mass of cylinder was defined asm=4.418× 1 0-6kg and the spring stiffness was defined ask=6.27× 10-3N/m,therefore, the natural frequency of the cylinder could be calculated asfn=6 Hz. The kinematic viscosity of fluid wasν=1× 1 0-5m2/s. In this study, a series of gap distance was considered from 0.5Dto 2.0Das well as the vertical distance between plates from 0.5Dto 1.2D. The reduced velocity (Ur=U/fn*D) was changed from 3.3 to 6 while the range of 198 ≤Re≤ 3 60 was considered. The twodimensional unsteady laminar flow model is applicated within the consideredRerange.

2. Numerical method

2.1 Governing equations and flow quantities

The flow dynamics are described by the incompressible Navier-Stokes equations.

whereuis the velocity of the fluid,pis the pressure of the fluid andρandνare the density and the kinematic viscosity of the fluid.

In this simulation, drag and lift force coefficients are used to quantify the hydrodynamic characteristics.The definitions are also given here:

whereFDandFLare the fluid forces exerted on the bodies, respectively, in the streamwise and transverse direction, and are calculated by performing an integration along the wall, involving both pressure and viscous effects.

The governing equation for the dynamic response of an elastically mounted cylinder which is simplified as a mass-spring system in the transverse direction reads as follows[19]

whereY,Y˙ andY˙ denote the displacement,velocity, and acceleration of the cylinder, respectively,ξis the structural damping ratio andMris the reduced mass,. The non-dimensional force componentCland reduced velocityUrhave been given in the previous description.

There are several approaches to resolve the moving boundary problem, such as the moving mesh method[20], immersed boundary method[21]and etc..The immersed boundary method is an effective choice to resolve the problem with large deformation and movement, but it is not very accurate at high Renolds number. In the present study, the interaction of the fluid and structure is investigated by the Arbitrary Lagrangian-Eulerian method[22]. Dynamic meshes are used to handle the moving boundaries of the vibration cylinder. Moving mesh in finite volume method is based on the integral form of the governing equation over an arbitrary moving volume V bounded by a closed surfaceS. For a general tensorial propertyφit states[23]

whereρis the density,nis the outward pointing unit normal vector on the boundary surface,uis the fluid velocity,usis the velocity of the boundary surface,φγis the diffusion coefficient andsφthe volume source/sink ofφ. Relationship between the rate of change of the volumeVand velocityusof the boundary surfaceSis defined by the space conservation law(SCL)[24]

The diffusivity model is used to determine how the points should be moved when solving the cell motion equation for each time step. In this study, inverse distance in the quality based methods was used to describe the mesh motion, where we specify a dynamic boundary and the diffusivity of the field is based on the inverse of the distance from the dynamic boundary.

The aerodynamic force is needed to solve the motion equation of elastically mounted cylinder, which can be obtained from the calculation of the flow field.However, the displacement of the cylinder resulted from the solution of the motion equation, in turn changes the boundary of the flow field and then, the hydrodynamic force on the cylinder. This is a typical fluid-structure interaction problem that can be easily solved with staggered algorithm(loose coupling), in which the flow field and the response of the structure are solved successively in a given time step.

2.2 Flow domain and boundary condition

Supposing a reference frame whose origin was fixed at the center of the cylinder before the vibration,the computational rectangle domain extended 40Ddownstream in the wake region, 16Dto the upstream boundary. The horizontal boundaries were located 20Dfrom the wake centerline to the upper/lower sides of the domain. The boundary condition used in the simulation were specified as follows: a Dirichlet-type boundary condition was used on the inlet boundary asu=Uandv=0, and on the outlet,a Neumann-type boundary condition was applied as?u/?x= 0 and ?v/?x=0. Symmetric free-slip walls were specified on the horizontal boundaries:?u/?y= 0 andv=0. On the surface of the cylinder and the splitter plates, a no-slip boundary condition was imposed, and the fluid velocity on the cylinder surface was determined by the kinematic condition of the cylinder motion while the velocity was set to be zero for the splitter plates(see Fig. 2).

Fig. 2 Computational domain and boundary conditions for the simulation

2.3 Grid system and validation study

To ensure the validation of the numerical simulation and the solution’s convergence, we have conducted grid independence test for VIV of the single cylinder on two grid system (fine grid and coarse grid whose generation parameters were tabulated in Table 1) with different mesh densities.The problem considered here is atRe= 150 and fixed structural parametersMr=2.00 andξ=0,which are the same with the research of Ahn and Kallinderis[25], Bao et al.[26]whose results are used here as reference data for comparison with our results.The time evolutions of dimensionless displacement of the cylinder are compared in Fig. 3. The max dimensionless displacements obtained in both grids were closed. The results demonstrate the spatial convergence on both grids. Therefore, all the simulations presented hereafter are based on the mesh generation parameters of the fine grid system (see Fig. 4 for the grid density near the cylinder).

Fig. 3 Time dependent dimensionless displacement of the cylinder calculated under different grid densities at Re=150

Fig. 4 (Color online) Close-up view of the structured mesh for the VIV of the single cylinder system

Table 1 parameters for the generation of two grid system and the validation study

3. VIV of a single cylinder

In this section, we present the calculated results of the VIV of the cylinder without any control strategies.The diameterDof the cylinder is 0.01 m and dimensionless massMrwas 5.93. The damping ratioξwas set to be zero to get the largest amplitude.The reduced velocity was extensively varied from 3.3 to 6.0, in order to determine its lock interval.

The max dimensionless displacement at each reduced velocity is plotted in Fig. 5(b) (Fig. 5(a) is the time history curve of the vibration displacement of the cylinder atUr=4.8, and the maximum value of the vibration response is plotted in the Fig. 5(b)). There is a branch of the higher amplitude region with maximum oscillation at 0.485 ≤Ymax/D≤0.536 while the reduced velocity varied from 4.5 to 5.0. In the outside of this region, the maximum cylinder oscillations decrease suddenly, their values are approximateYmax/D≈ 0 .07. Although different numerical methods were used in the present and previous computations,the result shows good agreement for the variation trends of the cylinder response with theUr.

Fig. 5 Response curves and comparison curves: (a) time history curve of the vibration displacement of the cylinder while reduced velocity is 4.8, (b) comparison of the maximum displacement at different reduced velocity

Figure 6 shows the vorticity contours forUr=3.3-6.0. AtUr=3.3, corresponding to the minimum amplitude of the vibration, the wake behind the cylinder presents a single-row vortex street which is composed of alternatively shedding vortices while the vortex intensity was low at this time. As theUrincreased to 4.5, the vortex street with high vorticity shed from the cylinder tends to move toward to its two sides, and finally a double-row vortex street is formed near the rear of the cylinder. whileUr≥ 4 .8, the wake pattern returns to the single-row vortex street behind the cylinder. Although the vortex distribution is similar asUr≥ 4 .8, there is a big difference in the way of vortex shedding. For the lager response interval, the vortex obviously shed from the surface of the cylinder to the upper or lower sides of the cylinder, and the vortex in this interval has lager transverse distribution.However, the vortex shed from the edge of the cylinder to the rear of the cylinder in the smaller response interval. In view of such results, it can be inferred that the large response of oscillation is caused by the strong velocity vector of the vortex shedding. The hydrodynamic force coefficients are plotted along in Fig. 9 with the results of the control system.

4. Flow over a vibration cylinder with two stationary splitter plates

4.1 The displacement response modification

The changes in the displacement response of the cylinder resulted from the wake disturbance due to the stationary splitter plates are shown in Fig. 7. The distanceSbetween the splitter plates and the cylinder was fixed at the different value in Figs.7(a)-7(d), respectively. The displacement responses corresponding to the different heightHbetween the two splitter plates were represented by different line type at the fixed gap spacings.

Apparently, due to the existence of the splitter plates, the dynamic response curve varies with changes in the configuration of plates. The response curves of control systems, while the gap spacing is 0.5D, were plotted in Fig. 7(a). The vibration of the cylinder in the lock interval was totally suppressed under all the control strategy whileS/D= 0.5. As the Ur increase to 5.5, the control system in which theH/D≥ 1 .1 cannot further reduce the amplitude of the cylinder. As theUrincreased to 5.8, the most control system had lost their effect except the case thatH/D= 0.7, 0.8. We have to admit that in the all control system the presence of the plates can’t reduce the amplitude of the cylinder at theUr=6.0. In general, the oscillation amplitude in the lock interval was reduced by more than 50%, and the oscillation amplitude at theUr=5.8, 6.0 did not exceed the amplitude in the lock interval without control.Consequently, we can get the conclusion that the dynamic response of the cylinder was suppressed due to the effect of the control system in which theS/D= 0.5 and the best control effect appeared inH/D= 0.7 or 0.8.

Fig. 6 (Color online) The vorticity contours corresponding to different reduced velocity Ur

As the gap spacing increase to 1.0D, the max oscillation response of the cylinder with the control system was similar to the case without the control system. The amplitude of the cylinder is not significantly suppressed. Although the control system can affect the amplitude of the cylinder in the lock interval well, the amplitude response out of the lock interval was also amplified with the presence of the plates.

As the gap spacing reached 1.5D, the amplitude of the cylinder cannot be suppressed with any control system. In view of the dimensionless displacement curve, it seems the existence of the plates may stretch the response interval of the cylinder, so that the lock interval is shifted to the right and became wider.Similarly, the results ofS/D= 2.0 shows that the plates just amplify the amplitude of the cylinder.

Accordingly, the closer the plates are to the tail edge of the cylinder, the better the control effect of the amplitude of the cylinder is. As the gap spacing increase, the effect of the plates on the cylinder amplitude is gradually weakened, and even the displacement of the cylinder would be amplified. In order to determine the optimal height of the plates at the gap spacing is 0.5D, more control height increments were considered in this paper. The dimensionless amplitude curves show that the optimal control system isH/D= 0.7 or 0.8 whileS/D= 0.5.

4.2 Hydrodynamic force characteristic

The flow modification caused by the imposed disturbance in the near wake can be quantitatively reflected in the variation of the hydrodynamic force coefficients. Three typical cases with a different combination of the height between the plates and the gap spacing are presented to illustrate the hydrodynamics mechanism of the flow filed. The temporal fluctuations of the lift and drag coefficients of the cylinder are shown in Fig. 8. As seen in Fig. 8(a), the cylinder’s drag is induced to fluctuate in the whole simulation process due to the disturbance. Apparently,the stabilization process varies with different flow statuses. In the beginning, all the drag undergoes sequential stages of first decreasing and then increasing.Furthermore, in the flow case II ( S / D =1.0 ,H/D= 0.5) and III (S/D= 1.5,H/D= 0.6), the drag undergoes a period of stable fluctuation and then increases to the saturated state, which are the characteristics of the recirculating bubble and the vortex shedding in the evolution process. However, in the flow case I (S/D= 0.5,H/D= 0.8), the unsteady drag keeps a significantly lower value with continuous fluctuation, which implies that the wake holds its stable feature consistently. In terms of lift coefficient,the case II (S/D= 1.0,H/D= 0.5) and III(S/D= 1.5,H/D= 0.6) have the same characteristics with the drag coefficient. Obviously, there is a delay of response in the case II compared with the case III regardless of the lift or drag coefficients. The reason is that the vortex shedding was postponed by the presence of the plates at this distance. The case ofS/D= 1.5,H/D= 0.8 identifies a lower modulation frequency in the drag and lift fluctuation,which is directly caused by the unsteady interaction of the wake and plates.

Fig. 7 (Color online) The displacement response modification at different gap spacing

Fig. 8 (Color online) Time series of the hydrodynamic force coefficients at different gap spacing and height ( =U r 4.8)

The statistical results of the hydrodynamic force coefficient, including the time-averaged and root mean square (rms) values, are presented in Fig. 9. It can be seen that the gap spacing between the cylinder and the plates has a greater influence than the height between the plates from the figures of the displacement response. Therefore, a set of control strategies under different gap spacings are selected to plot. In general, the drags on the cylinder increased with the increase of the gap spacing in terms of mean and rms value. Besides, in the range of larger gap spacing (S/D≥ 1 .0), the rms lift on the cylinder is not very sensitive with the gap spacing which is resulted from the unsteady nature of the flow at these ranges. But the smaller gap spacing has a significant influence, that is similar to the conclusions of Kawai[17]. In the higher spacing (/ 1.0)S D≥ , a sudden increase in the mean and rms. statistics of the hydrodynamic forces are observed for the cylinder in theUrrange of 4.5-5.0, except for the fluctuation of the drag in theS/D= 1.0 on the cylinder, which is associated with the wake transition in the gap flow region.

Fig. 9 (Color online) Statistical parameters of the hydrodynamic forces exerted on the cylinder

4.3 Vortex shedding frequency

It is meaningful to explore how the splitter plates influence the frequency characteristics of the vortex shedding. The ratio of the shedding frequency to the natural frequency of the cylinder (fn=6 Hz) will be compared here. The vortex shedding frequency (fv)is obtained by performing Fourier transform to the lift coefficients. As seen in Fig. 10, the time history curve of the VIV of the cylinder with no control system at theUr=4.8 are plotted in Fig. 10(a). The Fourier transform is used to obtain the frequency as shown in Fig. 10(b) and the peak was taken as the shedding frequency (fv) here.

Fig. 10 Representative results of lift coefficients (a) and its frequency domain information (b) of the cylinder without control systems (Ur=4.8)

The ratio of the vortex shedding frequency to the natural frequency of the cylinder is plotted in Fig. 11.In the same way, a set of control strategies under different gap spacings are selected to analyze. In view of the curve representing the cylinder without any control systems, the shedding frequency equals the natural frequency of the cylinder at the higher amplitude interval, which is consistent with the amplitude figures. This interval is called the lock-in interval. The rms value of the lift coefficients in the interval isn’t very big which is also confirmed in the Ref. [26], it is the lock-in of the frequency that leads to the high amplitude response of the cylinder.Additionally, outside the lock-in interval, the frequency ratio increase with the increase of the reduced velocity. While the gap spacing equals to 0.5D, the shedding frequency increased slowly with the reduced velocity until theUr=5.5. As theUr≥ 5 .8, the ratio of the shedding frequency to the natural frequency of the cylinder was closed to 1.0,which would lead to an increase of the amplitude in the displacement diagram, in fact, it’s true as seen in Fig. 7(a). And other control strategies not only have a frequency ratio closed to 1.0 in the lock-in interval,but also fluctuate around the lock-in state outside the lock-in interval. In consequence, the interaction of the vortex shedding from the trailing edge of the cylinder and the plates easily caused the lock-in phenomenon while the gap spacing exceeds 0.5D.

Fig. 11 (Color online) Statistical diagram of the ratio of shedding frequency to the natural frequency of the cylinder

4.4 Flow pattern modification

The conversion of the flow pattern caused by the wake disturbance due to the splitter plates is shown in Fig. 12. The wake dynamic modification is categorized into three regimes according to the difference of the vortex shedding and displacement of the cylinder. In the regime I, the contact of the shear layers is significantly delayed. The vortex shedding develops behind the splitter plates, and the oscillation strength is considerably reduced, while the wake behind the plates exhibits a single-row vortex street.Almost all cases ofS/D= 0.5 while the amplitude of the cylinder is lower and part cases ofS/D=1.0(H=0.5, 0.6 atUr=5.8) belong to the regime I.Figure 12(a) shows the feature of the flow regime I.As for the flow regime II, the cases in which dimensionless displacements of the cylinder varied from 0.2 to 0.3 are nearly at this region. The vortex that hasn’t been fully developed after the cylinder interacts with the splitter plates. Thus, the vortex in the far wake is alternately distributed in double rows as shown in Fig. 12(b). In the other cases, the flow patterns belong to regime III. The wake feature of regime III is shown in Fig. 12(c). A part of vortices which periodically shed from the upstream cylinder move into the gap between the splitter plates, and then the flow oscillation in the gap induces vortices shedding from the plates. Finally, the wake behind the plates exhibits a triple-row vortex street. The splitter plates fail to attenuate the vortex shedding from the cylinder in these gap distance ranges. This results that the oscillation amplitude of the cylinder cannot be suppressed, as shown in Fig. 7.

Fig. 12 (Color online) Instantaneous vorticity contours for the different flow regimes

As mentioned above, the flow regimes can be reflected in the displacement of the cylinder. It is meaningful to discuss the relationship between the forces exerted on the cylinder and the vorticity field.In order to reveal the mechanisms of the hydrodynamic force discussed above, the vorticity and the velocity vector figures of the same cases are selected to analyze. The contours of the vorticity and the velocity vector figures of three cases withS/D= 1.0,H/D= 0.5,S/D= 1.5,H/D=0.6 andS/D= 0.5,H/D= 0.8 are compared in Fig.13 (Ur=4.8). In the case ofS/D= 1.0,H/D=0.5, the vorticity fields showed in Fig. 13(a) belong to the flow regime II. Although the vortexes shedding from the cylinder are not completely developed between the cylinder and the plates, a jet field as shown in Fig. 13(b) is still generated at the trailing edge of the cylinder owing to this vorticity field. In this case, the flow field exerts a relatively higher oscillation force on the cylinder. The plates merely delay the increase of the dimensionless displacement of the cylinder with the reduced velocityUras shown in Fig. 7 for the plates locate the core of shedding vortexes. In the case of theS/D= 1.5,H/D= 0.6,the vorticity figure, more precisely Fig. 13(c), showed the features of the flow regime III. As the vortexes shedding from the cylinder are transferred downstream, the vortex streets are squeezed to two sides by the splitter plates. Thus, the wake width increases, and stronger velocity vector just like Fig. 13(d) was induced by this vorticity structure. The stronger jet field is, the higher hydrodynamic force coefficients and displacement response are just as we can see above. The presence of the jet field is detrimental to the vibration control of the cylinder, which would amplify the amplitude of the cylinder. However, in the case of theS/D= 0.5,H/D= 0.8, whose flow pattern belongs to flow regime I, the region of high vorticity behind the cylinder appears in form of a more streamlined shape compared with the regime II and III, and the vortex streets are developed after the plates. As a matter of fact, it can be seen in the Fig.13(f), the vorticity field does not induce a strong jet,and the velocity between the cylinder and the plates is relatively small and the direction is distributed along the streamwise direction. It explains why the control effect is more effective in this gap spacing.

These results show that the instability features of the structure wakes are strongly influenced by the flow disturbance, which was also confirmed in the Refs. [27-28]. The cylinder motion would be greatly changed by the different wake disturbances, and so was global wake instability. The performance of different control systems suggested that the zone of attenuating instability is mainly affected by the gap spacing between the cylinder and the plates.

5. Conclusions

The wake stabilization and the vibration control effect of two splitter plates distributed along the wake centerline in the near wake region of a circular cylinder is numerically investigated in the laminar flow regime. Three flow regimes (I, II, and III) are identified based on the modification of the wake dynamic characteristics. In regime I, the wake is significantly suppressed, and the flow oscillation is detected behind the plates. It shows that the presence of the stationary plates can delay the interact of the shear layers. There is a tendency for vortices shed from the cylinder in regime II that the vortex is not completely developed and the increase of the dimensionless displacement of the cylinder with the reduced velocity is delayed due to the existence of the plates. In flow regime III, the splitter plates are unable to stabilize the wake, evidenced by a fully developed vortex street in the gap.

Fig. 13 (Color online) The comparison of the flow fields of different flow regime ( Ur =4.8) : (a) and (b) vorticity contour and velocity vector of the case S/D =1.0 , H/D = 0.5 , (c) and (d) vorticity contour and velocity vector of the case S/D =1.5 , H/D = 0.6 , (e) and (f) vorticity contour and velocity vector of the case S/D = 0.5 , H/D = 0.8

The comprehensive parametric analyses show that the wake suppression and vibration control effect of the control system are enhanced with the decrease of the gap distance. The control effect of the cylinder is optimal while S / D = 0.5 , H / D = 0.8 . The displacement response is reduced by up to 91%(Ur=4.8). As the gap spacing increased larger than 1.0D, the dynamic response modes of the flow are mainly II and III, which cannot play a control role,and may amplify the response of the displacement.The shedding frequency of the control system whose gap distance≥1.0Dfluctuate around the natural frequency of the cylinder. This is caused by the interaction of the wake flow of the cylinder and the plates. The current work only consider a physical parameter of the cylinder, but the core mechanism, control of the vortex shedding position, is believed to be applicable to most vortex-induced vibration problems.

The performance of the wake control system is related to the reduced velocity. The instability of the wake and the amplitude of the cylinder are increased at a higher reduced velocity. But even at the maximum reduced velocity, the optimal control system can ensure that the amplitude response does not exceed the maximum amplitude of the cylinder without control system. It is also found that when the gap spacing increases, the lock-in interval of the cylinder may be enlarged.