Numerical simulation and experiment investigation on passive-suction-jet control of wind effect of two tandem cable models

Abstract

Two tandem cables are frequently employed as one group of hangers in a long-span suspension bridge; however, if they are close to each other, the mutual interaction between their flow fields is prone to produce large wind/wake-induced vibrations. In the present study, initially, a numerical simulation was conducted to investigate the interaction between two static tandem cable models with different spacing ratios, S_R (center-to-center longitudinal spacing divided by the cable diameter, i.e. L/D). Concurrently, the passive-suction-jet control method was employed to eliminate the interaction of these two tandem cables. Aerodynamic coefficients and time-averaged and instantaneous flow fields were used to evaluate the effectiveness of the passive-suction-jet control. Subsequently, the passive-suction-jet control method was employed in a wind tunnel experiment to manipulate the wind-induced vibrations of two elastically mounted cable models. The flow patterns of the controlled tandem cables were subdivided into three basic regimes in the present study. Furthermore, the aerodynamics force suppression mechanism was explained based on the flow patterns. Both the aerodynamic forces and vibration responses of the tandem cable models reduced significantly when S_R > S_Rc (critical spacing ratio). Particularly for S_R = 4.0, the lift fluctuation reduction of both the cable models was remarkable, the fluctuating lifts of the upstream and downstream cable models decreased by 93.3% and 72.1%, respectively, and the vortex-induced vibration responses decreased by 31.4% and 54.0% respectively. Furthermore, the wake-induced vibration responses of the tandem cable models could be completely suppressed when both were controlled using passive-suction-jet pipes.

Keywords

aerodynamic force flow pattern passive-suction-jet control tandem cable models vibration response

Introduction

Cables of suspension bridges are typical cylindrical and important force-bearing structures. However, owing to the low damping and large slenderness ratio of a cable structure, a single cable is prone to exhibit wind-induced vibrations under the action of an oncoming flow. In case of large bridge spans, two cables are frequently employed as one group of hangers in the suspension bridges. When the centerline of such two cables is parallel to the direction of the oncoming flow, the downstream cable is just immersed in the wake of the upstream cable; this is called as a tandem configuration. The flow fields of two tandem cylindrical structures contain complex mutual interactions, which increase the complexity of the aerodynamic forces, surface pressure distributions, and flow patterns of the structures (Zdravkovich, 1987; Zdravkovich and Pridden, 1977). Moreover, they induce vortex-induced vibrations (VIVs) and wake-induced vibrations (WIVs) of the structures (Assi et al., 2013; He et al., 2020). Therefore, it is essential to conduct flow control research on the flow fields and vibration responses of tandem cables.

For two same-diameter (D) tandem cylinders over the subcritical Reynolds number Re = 800–1.2 × 10⁵, numerous researchers have studied the aerodynamic characteristics (Alam et al., 2003; Igarashi, 1981; Kitagawa and Ohta, 2008; Ljungkrona et al., 1991; Xu and Zhou, 2004; Zdravkovich and Pridden, 1977), surface pressure distributions (Alam et al., 2003; Igarashi, 1981; Kitagawa and Ohta, 2008; Ljungkrona et al., 1991; Zdravkovich and Pridden, 1977), and flow field characteristics of static tandem cylinders (Alam et al., 2003; Igarashi, 1981; Kitagawa and Ohta, 2008; Lin et al., 2002; Ljungkrona et al., 1991; Zdravkovich, 1987; Zhou and Yiu, 2006) and classified a series of flow patterns (Igarashi, 1981; Xu and Zhou, 2004; Zdravkovich, 1987; Zhou and Yiu 2006). Different from the flow patterns of a single cylinder, which mainly depend on its Re, those of two tandem cylinders are largely affected by both Re and the spacing ratio, S_R (center-to-center longitudinal spacing divided by the cable diameter, i.e. L/D). Igarashi (1981) classified eight different flow patterns for two same-diameter static tandem cylinders in a subcritical region. Zdravkovich (1987) divided the flow pattern into three basic flow regimes: (1) For 1 < S_R < 1.2–1.8, the shear layers of the upstream cylinder directly roll up behind the downstream cylinder to form Karman vortices. (2) For 1.2–1.8 < S_R < 3.4–3.8, the shear layers detached from the upstream cylinder reattach to the surface of the downstream cylinder, and only the downstream cylinder forms the Karman vortices. (3) For S_R > 3.4–3.8, the shear layers detached from the upstream cylinder roll up in the gap and form alternating shedding Karman vortices, that is, a Karman vortex street is formed behind each cylinder. Xu and Zhou (2004) and Zhou and Yiu (2006) referred to the above three regimes as the extended-body, reattachment, and co-shedding regimes, respectively. Because the flow patterns of tandem cylinders are sensitive to Re, the spacing range corresponding to each flow regime defined by Zdravkovich (1987) is slightly different from those by Xu and Zhou (2004) and Zhou and Yiu (2006). The spacing ratio at which the upstream cylinder just forms an alternating shedding vortex is called as the critical spacing ratio (S_Rc), which mainly depends on Re. On the flow regime transition from the attachment regime to the co-shedding regime, the mean and fluctuating drag forces and fluctuating lift forces of two tandem cylinders increase significantly. This occurs mainly because the vortices shedding from the upstream cylinder impinge on the downstream cylinder and merge with the vortices formed by the downstream cylinder, causing the flow field around the downstream cylinder to be more unstable (Alam et al., 2003; Dehkordi et al., 2011; Kitagawa and Ohta, 2008).

Cylindrical structures, for example, cables of suspension bridges, are adversely affected by the lift fluctuation caused by the alternating vortex shedding. Researchers have proposed numerous schemes to control the flow field around a cylinder to suppress the alternating vortex shedding and reduce the aerodynamic forces acting on the cylinder. Depending on the energy input or not, the flow control methods can be classified as active and passive. Active control methods include traveling wave walls (Wu et al., 2007; Xu et al., 2014), affiliated control cylinders (Dipankar et al., 2007; Zhu et al., 2015), active suction or blowing (Chen et al., 2014; Kim and Choi, 2005), and synthetic jets (Feng and Wang, 2010; Tensi et al., 2002). These active control methods can achieve good control effects; however, they have the disadvantages of complex systems and high costs. Contrastingly, passive control methods mainly realize flow control by changing the cross-section or surface characteristics of a cylinder; they have the advantages of simple device and low cost, and can achieve a good control effectiveness under the designed conditions. Some examples are as follows. The sharp peaks of a U-type groove can break the longitudinal vortices into smaller ones (Lim and Lee, 2002). A wavy cylinder or a cylinder with bumps in a spiral pattern are effective in reducing the drag and suppressing the VIV; however, a wavy cylinder is only suitable for unidirectional flows (Owen et al., 2001). The non-uniform surface of a bionic cylinder based on harbor seal vibrissa weakens the spanwise correlation in the near wake and effectively reduces the aerodynamic forces (Chen et al., 2018). Baek and Karniadakis (2009) set a slot inside a 2-D cylinder, and thus, formed a flow-communicating channel between the front and rear stagnation points. The jet flow blown out from the slot interacts with the wake flow, and as the slot width increases, it shows two control mechanisms: weakening or detuning of the alternating vortex shedding. Gao et al. (2017) and Zhu et al. (2019) separately explored the control effect for a three-dimensional (3-D) slotted cylinder at subcritical Re and the influence of different types of slots for a 2-D slotted cylinder at low Re.

Chen et al. (2015) designed an innovative passive-suction-jet control method, a modification of the internal slotting method. Hollow pipes were installed on the outer surface of a cylinder, and some holes were evenly distributed on the outer surface of the pipes, thereby forming a flow-communicating channel inside them. The jet flow blown out from pipes interacts with the wake flow and pushes the wake vortex structures downstream, thus transforming the wake vortex to a symmetric shedding mode. Furthermore, studies have found that a passive-suction-jet control can not only effectively manipulate the flow field around a static circular cylinder (Chen et al., 2015, 2019) but also suppress its VIV response (Chen et al. 2020). Previous studies proved that the passive-suction-jet pipe control method is effective in controlling the flow and vibration of a single cable segment model. However, owing to the strong mutual interference between tandem cables, the flow field around them is relatively complex; hence, the chief aim of the present study is verifying the passive-suction-jet control still has a preferable control effect on two tandem cables.

In the present study, initially a numerical simulation was utilized to investigate the interaction between two static tandem cable segment models with different spacing ratios S_R. Concurrently, the passive-suction-jet control method was employed to eliminate the interaction of these two tandem cables. Subsequently, the passive-suction-jet control method was employed in a wind tunnel experiment to manipulate the wind-induced vibrations of two elastically mounted cable models. The structure of this paper is as follows: The numerical model, validation of the flow around a single cable model, and aerodynamic and flow field characteristics of the static tandem cable models with and without control are presented in section “Numerical simulation.” The experiment model, validation of the VIVs of a single cable model and the vibration responses of two tandem cable models with and without control are discussed in section “Wind tunnel experiments.” The final section provides a summary.

Numerical simulation

Numerical model and validation

Validation of flow around single cable model

A numerical simulation of the flow around a single cable model is first taken as an example to verify the grid independence and the accuracy of the numerical method. The cable diameter is set as 100 mm, the fluid density is 1.225 kg/m³, the dynamic viscosity coefficient is 1.789 × 10⁻⁵ kg/(m·s), the oncoming wind speed is 2.921 m/s, and the corresponding Re is 2.0 × 10⁴.

Ansys ICEM CFD 19.0 is adopted for the meshing of the computational domain, and Ansys Fluent 19.0 for the numerical calculation. The computational domain and the boundary conditions are shown in Figure 1(a). The calculation domain is set as 40D × 20D × 4D, the center of the cable is the origin of the coordinates, and the center of the cable is 10D and 30D from the entrance and the exit, respectively. The inlet is the velocity inlet boundary condition, the outlet is the pressure outlet boundary condition, and the spanwise walls are set as the periodic boundary condition. The upper and lower walls are zero-shear slip walls, that is, the symmetric boundary condition, and the surface of the cable is a no-slip wall.

Figure 1.

Schematics of computational domain and grid partition of flow around single cable: (a) computational domain and boundary conditions and (b) mesh of xoy plane.

The computational domain is divided into structured grids, and O-shaped meshes are used around the cable, as presented in Figure 1(b). The three mesh types of coarse, standard, and fine are divided from sparse to dense for the flow around the static single cable. The mesh parameters are listed in Table 1, where N is the total number of grids, N_c is the number of nodes in the circumferential direction of the cable, and R_in is the grid height growth rate near the cable and at the wake of the cable. In addition, R_out is the grid height growth rate of the remaining zones, △y/D is the ratio of the normal distance from the wall to the diameter of the cable, and △z/D is the ratio of the grid height in the spanwise direction to the diameter of the cable. The maximum aspect ratios of all the three meshes are maintained at 62.5. Compared to the coarse mesh, the standard mesh further refines the grids in the spanwise direction, near the cable, and at the wake. The cable is divided into 192 parts in the circumferential direction and 80 parts in the spanwise direction. R_in and R_out are set as 1.04 and 1.05, respectively. To solve the viscous sublayer near the wall, the normal distance from the wall is 0.0008D. The total grid number of the standard mesh is 3,280,000.

Table 1.

Mesh parameters of flow around fixed single cable.

Mesh	N	N_c	R _in	R _out	△y/D	△z/D
Coarse	1,910,000	160	1.05	1.05	0.001	0.0625
Standard	3,280,000	192	1.04	1.05	0.0008	0.05
Fine	3,680,000	192	1.03	1.05	0.0008	0.05

Before the transient calculation, 8000 steps of the steady-state calculation using the Reynolds stress model (RSM) are adopted as the initial field. The residuals of the continuity and the three velocity components are set as 10⁻⁶. The residuals of the remaining indicators are set as 10^-5, and the transient calculation time step is 0.0002 s. The transient statistics are about 40 cycles of the lift coefficient time histories after the numerical solution is stable. As the main parameters of the aerodynamic characteristics, the lift coefficients and drag coefficients are calculated as follows:

C_{d} = \frac{F_{d}}{0.5 ρ U^{2} Dl}

(1)

C_{l} = \frac{F_{l}}{0.5 ρ U^{2} Dl}

(2)

where F_d and F_l are the total drag and the lift forces acting on the model, respectively, D is the outer diameter of the model, and l is the length of the model.

Based on the drag and lift coefficient time histories, the mean drag coefficients, C_d_-mean, and the RMS (root-mean-square) lift coefficients, C_l_-rms, of the three meshes can be obtained. By the Fourier transform of the lift coefficient time histories, the vortex shedding frequency (f_v) can be calculated as 6.027 Hz, which corresponds to the Strouhal number, St = f_vD/U = 0.206. The present aerodynamic characteristics of a fixed single cable are compared to previous results, as summarized in Table 2. The aerodynamic results are basically consistent with the previous results; however, the RMS lift coefficient is slightly larger, which may be related to the turbulence intensity of the oncoming flow and the slenderness ratio of the model.

Table 2.

Comparison of aerodynamic results of flow around fixed single cable.

Mesh	Re	I_u	C_d _-mean	C_l _-rms	St
Coarse	2 × 10⁴	1.0%	1.275	0.639	0.206
Standard			1.251	0.607
Fine			1.221	0.562
Moeller and Leehey (1984)	1.93 × 10⁴	0.9%	1.35	0.60	—
West and Apelt (1993)	2.8 × 10⁴	1.25%	1.17	0.59	—
Norberg (2003)	2.03 × 10⁴	0.06%	—	0.47	0.19
Chen et al. (2014)	3 × 10⁴	0.8%	1.24	—	0.21
Chen et al. (2015)	4.16 × 10⁴	0.36%	1.28	0.47	0.21

In addition, the dimensionless normal distance from the wall, y⁺ = y u_τ /ν, obtained from the standard mesh is 1.1, where y is the normal distance from the wall, u_τ is the friction velocity, and ν is the kinematic viscosity of the fluid, which meets the requirement of the solution of the viscous sublayer. Based on the points where the average shear stress in the downstream direction of the cylindrical surface is zero, the separation point is determined to be 85°, which is very close to the empirical value of 82°. Figure 2 shows the time-averaged flow field of the flow around the single cable. The vortex formation length is defined as the distance from the center of the cable to the farthest point of the recirculation zone. It is 1.2D in the numerical simulation, which is very close to the vortex formation length of approximately 1.3D obtained from experiments (Chen et al., 2014, 2015). From the above discussion, the accuracy of the numerical method and the grid independence are fully verified, and the grid partition method of the standard mesh and the numerical method are adopted for the numerical simulation of two tandem cables.

Figure 2.

Time-averaged flow field of flow around single cable: (a) on middle section (z = 0.2 m) and (b) on spanwise section (y = 0).

Flow control scheme of tandem cable models

Based on the passive-suction-jet control on the flow structure around a single cylinder conducted by Chen et al. (2015, 2019, 2020), in this study, the control parameters of the passive-suction-jet pipes are set as follows. The radial thickness and height of the pipes are 0.05D and 0.8D, respectively, and 24 holes are evenly distributed on the outer surface of the pipes, as shown in Figure 3. The center angle corresponding to the center of the adjacent holes is 15°, and that corresponding to the circumferential width of the holes is 7.5°. Adjacent pipes are densely distributed along the spanwise direction.

Figure 3.

Schematics of passive-suction-jet pipe in numerical simulation: (a) cutaway view and (b) front elevation.

Six S_R values of 1.5, 2.0, 3.0, 4.0, 5.0, and 6.0 are chosen to present different flow patterns of the two tandem cables. Based on the range of S_R, the calculation domain is set as 45D × 20D × 4D, as shown in Figure 4(a). C1 and C2 represent the upstream and downstream cable models, respectively, and the coordinate origin is located at the center of the upstream cable model. The positive direction of the drag coefficient is consistent with the direction of the oncoming flow, pointing to the positive direction of the x-axis. The positive direction of the lift coefficient is along the positive direction of the y-axis.

To explore the influence of a passive-suction-jet control on the tandem cables and its control mechanism, two different control schemes are designed, as shown in Figure 4(b). The baseline case with no control of the two tandem cables is designed for comparison, and the controlled case is in which both cables are controlled by passive-suction-jet pipes.

Figure 4.

Schematics of computational domain and control schemes of flow around two tandem cables: (a) computational domain and (b) control schemes.

The grid partition method of the flow around the two tandem cables is the same as the standard mesh of the single cable. Multiple O-shaped divisions are performed around the cables, and the total number of grids at various S_R is approximately 6,000,000. Taking the grid at S_R = 2.0 as an example, the schematics of the grid partition are displayed in Figure 5, where P1 and P2 represent the passive-suction-jet pipes of the upstream and downstream cables, respectively. It is should be noted that, the same mesh is used at the identical S_R in the numerical simulation for the baseline and controlled cases. The passive-suction-jet pipes adopt no-slip wall condition as the controlled case. Contrastingly, the pipes are set as the interior boundary condition as the baseline case, thus the fluid can pass through the pipes freely. Other boundary conditions are the same as the settings of the single cable.

Figure 5.

Schematics of grid partition of flow around two tandem cables: (a) cutaway view and (b) close-up of region near two tandem cables.

Control effectiveness of flow field around tandem cable models

Aerodynamic forces

The time histories of the aerodynamic coefficients of the two tandem cables at various S_R are shown in Figure 6. The definition of the aerodynamic coefficients for tandem cables can refer to equations (1) and (2), and it should be noted that the aerodynamic forces of the controlled cable models are taken as the sum of those of the cylindrical surface and the internal and external surfaces of the passive-suction-jet pipes. Moreover, the characteristic size of the controlled cable is the outer diameter of the pipes.

Figure 6.

Aerodynamic coefficient time histories of tandem cables with and without control at various S_R: (a) S_R = 1.5, (b) S_R = 2.0, (c) S_R = 3.0, (d) S_R = 4.0, (e) S_R = 5.0, and (f) S_R = 6.0.

Overall, the mean drag of the upstream cable is greater than that of the downstream cable for the baseline and controlled cases, whereas the fluctuating lift of the upstream cable is smaller than that of the downstream cable. When S_R increases to 4.0, the mean drag and the fluctuations of the drag and lift forces of the two tandem cables increase markedly in the baseline case. Particularly, when S_R is less than 4.0, the mean drag coefficient of the downstream cable is negative because of the shielding effect of the upstream cable, and it becomes positive when S_R increases to 4.0. This suggests that the flow pattern at this spacing is changed from the attachment regime to the co-shedding regime.

In the controlled case, the fluctuating lift and the mean drag of the downstream cable also become large at S_R = 5.0, whereas the drag coefficient of the upstream cable is stable at 0.8–1.0 without significant fluctuation, and the fluctuating lift is close to zero.

The mean drag coefficient, C_d_-mean, and the RMS lift coefficient, C_l_-rms, are obtained from the statistical analysis of the time histories of the aerodynamic forces. The comparison of the uncontrolled tandem cables and the previous results is shown in Figure 7. C_d_-mean-1 and C_d_-mean-2 represent the mean drag coefficients of the upstream and downstream cables, respectively; C_l_-rms-1 and C_l_-rms-2 are the corresponding RMS lift coefficients of the upstream and downstream cables, respectively. The change trend of the aerodynamic forces of the uncontrolled tandem cables with the increase in S_R is consistent with the previous results, and S_Rc is within the range of 3.0–4.0, which is close to S_Rc = 4.0 in the study by Alam et al. (2003) and S_Rc = 3.25 in that by Kitagawa and Ohta (2008). Therefore, it is proven that the calculated results of the uncontrolled tandem cables are in good agreement with the reference results.

Figure 7.

Compare aerodynamic coefficients of two uncontrolled tandem cables with previous results: (a) mean drag coefficient of upstream cable, (b) mean drag coefficient of downstream cable, (c) RMS lift coefficient of upstream cable, and (d) RMS lift coefficient of downstream cable.

The aerodynamic results of the baseline and controlled cases at different S_R are presented in Figure 8. The black dash line represents the calculation results of a static single cable. In the baseline case, the mean drag and fluctuations of the drag and lift forces of the two tandem cables have a significant jumping phenomenon at S_R = 4.0. Particularly, the RMS lift coefficient of the downstream cable is markedly larger than that of a single cable at S_R ≥ 4.0 because of the wake vortex interference of the upstream cable. In the controlled case, the aerodynamic forces of the upstream cable are effectively suppressed. What is more, only the lift and drag of the downstream cable present a jumping phenomenon at S_R = 5.0, which is delayed by 1.0D compared to the baseline case. Moreover, the fluctuating lift of the downstream cable is significantly reduced.

Figure 8.

Compare aerodynamic coefficients of tandem cables with and without control at various S_R: (a) mean drag coefficients of tandem cables and (b) RMS lift coefficients of tandem cables.

Based on S_R, the aerodynamic control effects of the tandem cables with passive-suction-jet pipe control can be broadly classified into three categories: (1) When S_R = 1.5, the fluctuating lifts of the upstream and downstream cables are reduced by 80.1% and 22.3% compared to those of the baseline case. (2) When S_R = 2.0 and 3.0, the aerodynamic forces of the tandem cables in the controlled case are similar to those in the baseline case. (3) When S_R = 4.0, 5.0, and 6.0, S_Rc corresponding to the aerodynamic jumping phenomenon of the downstream cable in the controlled case increases, and the fluctuating lift of the tandem cables is effectively suppressed. In the controlled cases, the reduction in the aerodynamic forces of the tandem cables is the largest at S_R = 4.0, the mean drag and fluctuating lift of the upstream cable are reduced by 32.7% and 93.3%, respectively. Moreover, those of the downstream cable are reduced by 96.7% and 72.1%, respectively. Owing to the jumping phenomenon of the aerodynamic forces of the downstream cable, the control effectiveness on the aerodynamic force is weakened at S_R = 5.0 and 6.0. When S_R = 6.0, the mean drag and the fluctuating lift of the upstream cable are decreased by 24.6% and 94.7%, respectively. The mean drag of the downstream cable is close to that of the baseline case, and the fluctuating lift is reduced by 19.1%. Essentially, at the larger spacing ratios, that is, S_R = 4.0, 5.0, and 6.0, the flow control effectiveness is governed by the passive jetting flux, which depends on the oncoming velocity and the passive suction flux. The oncoming mean speed of the upstream cable is higher than that of the downstream one. Therefore, the control effectiveness of the aerodynamic forces of the upstream cable is larger than that of the downstream one at larger spacing ratios.

Flow pattern

Figures 9 and 10 present the time-averaged flow fields on the middle section (z = 0.2 m) and the spanwise section (y = 0) for the baseline and controlled cases. The cloud picture displays the standardized streamwise fluctuating velocity, u^*_x,rms, which is defined as the ratio of the streamwise fluctuating velocity to the oncoming wind speed. The black lines in the figures are the time-averaged streamlines of the flow fields around the cables. In the baseline case, the shear layers detach from the upstream cable to reattach to the surface of the downstream cable at S_R < 4.0, that is, the reattachment regime. At S_R ≥ 4.0, both the upstream and downstream cables form alternating shedding vortices in their wake region, that is, the co-shedding regime.

Figure 9.

Time-averaged flow field on middle section of flow around two tandem cables: (a) baseline case and (b) controlled case.

Figure 10.

Time-averaged flow field on spanwise section of flow around two tandem cables: (a) baseline case and (b) controlled case.

According to the classification of the three typical flow regimes of extended-body, reattachment, and co-shedding regimes, the flow patterns of the controlled case are classified as follows: (1) When S_R = 1.5, the shear layers of the upstream cable fall off directly behind the downstream cable, which is defined as Regime I. From the spanwise flow field, small-scale transverse vortices are formed between adjacent pipes in the gap, and the streamwise fluctuating velocity around the tandem cables is significantly reduced. (2) When S_R = 2.0–4.0, the jet flow blown out from the upstream pipe rolls up to form a vortex pair and the shear layers detached from upstream cable reattach to the surface of the downstream cable, which is defined as Regime II. Moreover, the vortices in the gap present strong 3-D characteristics in the controlled case. (3) When S_R = 5.0 and 6.0, the upstream cable generates alternating shedding vortices behind the vortex pair, which is defined as Regime III. Moreover, the vortex formation lengths of the upstream cable are 2.8D at both S_R = 5.0 and 6.0, which are more than twice of those in the baseline case. Owing to the influence of the wake of the upstream cable, only one pair of vortices is formed in the wake of the downstream cable.

Based on the classified three regimes, four typical S_R of 1.5, 3.0, 4.0, and 5.0 are selected for the analysis. The control mechanism of the passive-suction-jet method is explored by analyzing three instantaneous vorticity distributions corresponding to the minimum, zero, and maximum lift coefficients in a lift cycle of the downstream cable (i.e. t_min, t₀, and t_max), as shown in Figure 11. The distances in the x and y directions are expressed as dimensionless lengths X/D and Y/D, respectively. The vorticity (ω) is defined as the curl of the fluid velocity field, which characterizes the strength and direction of the vortex, and its component in the xoy plane, ω_z, is calculated as follows:

ω_{z} = \frac{\partial u_{y}}{\partial x} - \frac{\partial u_{x}}{\partial y}

(3)

where x and y are the coordinates and u_x and u_y are the components of the flow field velocity in the x and y directions, respectively.

Figure 11.

Instantaneous vorticity on middle section in baseline and controlled cases at S_R = 1.5, 3.0, 4.0, and 5.0: (a) t_min, (b) t₀, and (c) t_max.

As presented in Figure 11, the vortex shedding process of the upstream cable in the baseline case is adversely affected by the downstream cable at S_R = 1.5. The shear layers of the upstream cable reattach to the surface of the downstream cable, and the downstream cable forms alternating shedding vortices. Thus, the flow pattern belongs to the reattachment regime. In the controlled case, the oncoming flow passes through the flow channel of the pipe and forms a jet flow, which subsequently rolls up in the gap to form a series of small-scale vortices. It can be seen that as they move downstream, the small-scale vortices at the gap develop and merge into the wake vortex of the downstream cable. Clearly, the shear layers of the upstream cable do not reattach to the surface of the downstream cable, and the flow pattern is similar to the flow around a single bluff body, which is defined as Regime I. The jet flow pushes the shear layers of the upstream cable to both sides, so that the alternate force acting on the upstream cable is clearly weakened. Moreover, the periodic force acting on the downstream cable originated by the small-scale vortices, is not significant, comparing with that generated by the shear layers. Therefore, the fluctuating lifts of the tandem cables are effectively suppressed.

When S_R = 3.0, the flow pattern in the baseline case belongs to the reattachment regime, and the flow pattern in the controlled case is similar to that in the baseline case. The jet flow blown out from the pipe forms a vortex pair, and the shear layers of the upstream cable still reattach to the downstream cable. The flow pattern is defined as Regime II. However, the vortex pair increases the vorticity behind the upstream cable and pushes the vortex movement in the gap closer to the downstream cable, which enhances the interference on the downstream cable. Therefore, the aerodynamic control effect of the tandem cables is similar to that of the baseline case at S_R = 3.0.

At S_R = 4.0, both the cables in the baseline case form Karman vortices, which belongs to the co-shedding regime. The vortex shedding from the upstream cable impinges on the downstream cable and merges with the vortex formed by the downstream cable; thus, the aerodynamic forces of the downstream cable increase significantly. However, the jet flow forms a vortex pair directly behind the upstream cable, pushing the original vortex formation region downstream in the controlled case. Because the vortex formation region in the gap is reduced, the alternately shedding vortices cannot be formed, thus behaving as Regime II. Because there is no impingement and amalgamation of the vortices, the aerodynamic forces of the tandem cables are sharply reduced.

When S_R = 5.0, the flow pattern of the tandem cables in the baseline case belongs to the co-shedding regime. However, the flow pattern in controlled case behaves as Regime III, which is completely different from that of S_R = 4.0. The jet flow blown out from the pipe of the upstream cable forms a vortex pair in the near wake, and behind this vortex pair, the shear layers of the upstream cable roll up and form alternating shedding vortices. The jet flow not only increases the vortex formation length of the upstream cable but also reduces the interference of alternate vortices on the downstream cable to a certain extent. The alternate vortices of the upstream cable impinge with the downstream cable and merge with the vortex formation process of the downstream cable.

Wind tunnel experiments

Experiment model and validation

Control scheme and experiment model

The wind-induced vibrations of two tandem cables will not only produce VIVs in the tandem cables excited by their vortex shedding but will also generate WIVs in the downstream cable caused by the interaction with the wake of the upstream cable (Assi et al., 2013). For further investigating the effectiveness of the passive-suction-jet flow control method on the suppression of the VIV and WIV responses of two tandem cables, wind tunnel experiments of two elastically mounted cable models were conducted. The vibrations in the cross-flow direction were measured to evaluate the control effectiveness. Based on the conclusion of section “Numerical simulation,”S_Rc is between 3.0 and 4.0 at Re of 2.0 × 10⁴. Therefore, within the approximate Reynolds number range, three S_R of 3.0, 4.0, and 5.0 were chosen to conduct the wind-induced vibration experiments, which correspond to the common S_R of one group of tandem hangers of suspension bridges. Similarly, a baseline case without the control of the tandem cable models was designed for comparison, and subsequently, the controlled case in which both the tandem cable models are densely arranged pipes was used for the flow control analysis. The schematic of the vibration model is shown in Figure 12(a).

A rigid cable model is selected to densely install passive-suction-jet pipes on the outer surface of a hollow acrylic tube with D = 75 mm. The total length of the model is l = 720 mm. Each pipe is formed by 3-D printing technology, and the wall thickness of each pipe is 1 mm. The control parameters of the pipes are the same as those in the numerical simulation, and the width of the flow channel inside the pipes is approximately 0.05D; thus, the corresponding outer diameter of the pipes is D₁ = 86.4 mm. The detailed dimensions of a pipe are shown in Figure 12(b), and the schematic of the controlled cable models is presented in Figure 12(c). To unify the weights of the cable models, the baseline case is realized by closing the holes of the pipes; therefore, the diameter of the vibration model is uniformly taken as D₁.

Figure 12.

Schematics of vibration model of tandem cables in experiment: (a) vibration model, (b) dimensions of each pipe, and (c) controlled cable models.

Experiment setup

The experimental measurements were conducted in a wind tunnel (SMC-WT2) affiliated with the Wind Tunnel and Wave Flume (WTWF) Joint Laboratory of the Harbin Institute of Technology, P.R. China. The closed-circuit wind tunnel has a test section that is 800 mm wide and 1200 mm high. Each rigid cable model with end plates is fixed horizontally with a threaded rod of a 10-mm diameter, and four springs are installed on two ends of the threaded rod along the cross-flow direction. Acceleration sensors are connected to the threaded rod to capture the vibration acceleration of the cable model. The wind speed range of the vibration experiments is 2.0–10.0 m/s, and the Cobra Probe is used to statistically measure the oncoming wind speed during the acceleration collection process.

Validation of VIVs of single cable model

Based on the free vibration experiment of the single cable model, the inherent characteristics, such as the damping ratio, ξ, and the natural frequency, f_n, of the cable model were obtained. The total mass of each cable model is 1.91 kg, and the mass ratio can be calculated as m* = m/(πρD₁²l/4) = 369.36. The damping ratio and the natural frequency of the two tandem cable models are similar, which are close to 0.10% and 7.50 Hz, respectively. The corresponding Scruton number and reduced damping parameter are Sc = 0.5π (m*ξ)=0.58 and S_G=8π²S_t²ξm* =1.17, respectively.

Following the VIV experiment of the single cable model to verify the accuracy of the results, the vibration experiments of the controlled single cable model were conducted for initially exploring the effect of the passive-suction-jet control on the VIVs of the single cable model.

Figure 13 shows the comparison of the peak vibration amplitude of the uncontrolled single cable model and the previous results of Skop and Balasubramanian (1997). In the figure, 2A_max/D₁ denotes the ratio of the maximum vibration amplitude (2A_max) to the outer diameter of the cable model (D₁). It is clear that the results are very consistent with the previous results.

Figure 13.

Compare peak vibration amplitude of uncontrolled single cable model with previous results in Skop and Balasubramanian (1997).

The vibration responses of the single cable model with and without control are presented in Figure 14. The corresponding vibration amplitudes at the reduced velocity of 4.5–8.5 are displayed in Figure 14(a), where the expression for the reduced velocity (U*) is U* = U/(f_nD₁). The results indicate that the uncontrolled cable model exhibits VIVs within the reduced velocity range of 5.0–8.1, and the peak vibration amplitude of the uncontrolled cable model (i.e. 2A_max/D₁ = 0.50) occurs at the reduced velocity of 6.2. For the controlled case, the peak vibration amplitude of the controlled cable model is reduced by 80%, and the VIV is exhibited in the reduced velocity range of 5.2–7.4, which is reduced slightly compared to that of the baseline case. Figure 14(b) and (c) show the time histories and frequency spectra of the cable models with and without control at the reduced velocity of 6.2. It can be seen that the frequency amplitude of the controlled cable model is prominently decreased; therefore, the periodic vortex shedding is largely suppressed.

Figure 14.

Vibration responses of single cable mdoel with and without control: (a) vibration amplitudes of single cable, (b) time histories of single cable at reduced velocity of 6.2, and (c) frequency spectra of single cable at reduced velocity of 6.2.

Control effectiveness of wind-induced vibrations of tandem cable models

For the baseline and controlled cases, the vibration responses of the tandem cable models at S_R = 3.0, 4.0, and 5.0 are presented in Figure 15. Figures 16 –18 display the time histories and frequency spectra of the cable models at the reduced velocity corresponding to the peak vibration amplitude of the cable models in the baseline cases at S_R = 3.0, 4.0, and 5.0, respectively.

Figure 15.

Vibration responses of tandem cables with and without control at various S_R: (a) S_R = 3.0, (b) S_R = 4.0, and (c) S_R = 5.0.

Figure 16.

Time histories and frequency spectra of tandem cable models at S_R = 3.0: (a) time histories and frequency spectra of upstream cable at reduced velocity of 6.3, (b) time histories and frequency spectra of downstream cable at reduced velocity of 7.0.

Figure 17.

Time histories and frequency spectra of tandem cable models at S_R = 4.0: (a) time histories and frequency spectra of upstream cable at reduced velocity of 6.4 and (b) time histories and frequency spectra of downstream cable at reduced velocity of 6.6.

Figure 18.

Time histories and frequency spectra of tandem cable models at S_R = 5.0: (a) time histories and frequency spectra of upstream cable at reduced velocity of 6.7 and (b) time histories and frequency spectra of downstream cable at reduced velocity of 6.9.

For the baseline case, the VIV response distribution of the upstream cable model at S_R = 3.0 is broadly the same as that of the single cable model, and its relative peak vibration amplitude is 0.63 at the reduced velocity of 6.3. With the increase in S_R, the peak vibration amplitude dramatically decreases, and the relative peak vibration amplitude is 0.26 at S_R = 5.0, which is 48% smaller than that of the single cable model. Furthermore, as S_R increases, the wind speed range in which VIVs occur in the downstream cable model increases significantly; however, its peak vibration amplitude is almost equivalent to that of the single cable model. In addition, the downstream cable model presents a clear WIV phenomenon at S_R = 3.0 and 5.0 when the reduced wind speed exceeds 11.5 and 9.9, respectively, and the growth rate of the WIV response at S_R = 3.0 is larger than that at S_R = 5.0. Particularly, there is no WIV phenomenon in the downstream cable model at S_R = 4.0, which may be ascribed to the development of vortex impingement flow pattern (Sumner et al., 2000). Based on the conclusion of the present numerical simulation, the critical spacing ratio that the upstream cable just forms the Karman vortices in the gap is close to 4.0 in the vibration experiment. However, the critical spacing ratio is 2.5 in the study of Assi et al. (2010), and this may explain why the vibration response of downstream cable is visibly different from the results of Assi et al. (2010) at S_R = 4.0.

The WIV responses are completely suppressed when both the tandem cable models are controlled by passive-suction-jet pipes. According to Figures 16 –18, the frequency amplitudes of the tandem cable models are dramatically reduced, that is, the periodic vortex shedding is also prominently weakened. Thus, both the wind speed ranges in which the VIVs occur in the tandem cables and the VIV responses of the tandem cables are significantly reduced in the controlled case. The peak vibration amplitudes of the tandem cables at S_R = 3.0 are reduced by 57.1% compared to those of the baseline case. The peak vibration amplitudes of the upstream and downstream cables at S_R = 4.0 and 5.0 are reduced by more than 31.4% and 54.0%, respectively.

Conclusion

In the present study, the passive-suction-jet control scheme was employed to decrease the aerodynamic forces on two tandem cable models and manipulate the flow structures and wind-induced vibrations based on a numerical simulation and a wind tunnel experiment. The following conclusions were obtained:

(1) Except for S_R = 2.0 and 3.0, the fluctuating lifts of the controlled tandem cables were significantly suppressed. Particularly when S_R = 4.0, the lift fluctuation reduction of the two cable models was remarkable, and the fluctuating lifts of the upstream and downstream cables were reduced by 93.3% and 72.1%, respectively.

(2) The flow patterns of the controlled tandem cable models at various S_R were quite different from those of the uncontrolled tandem cable models because of the interaction effect of the jet flow blown out from the pipes with the main flow. Three flow regimes were defined to clarify the flow patterns of the controlled tandem cables at different S_R.

(3) The results of the wind-induced vibration experiment suggested that both the tandem cable models with densely arranged pipe control can completely eliminate the WIVs and effectively manipulate the VIV responses of the tandem cables. When S_R = 3.0, 4.0, and 5.0, the peak vibration amplitudes of the upstream and downstream cables were reduced by more than 31.4% and 54.0%, respectively.

(4) The aerodynamics force suppression mechanism was explained from the perspective of the flow regimes: (a) At a small S_R, the jet flow forms a series of small-scale vortices in the gap, which move downstream and merge into the wake vortex of the downstream cable. Thus, the periodic force of the small-scale vortices acting on the downstream cable is weakened. (b) At medium S_R, the jet flow forms a vortex pair, and the shear layers of the upstream cable still reattach to the downstream cable. Moreover, the vortex pair pushes the vortex movement in the gap closer to the downstream cable and enhances the interference on the downstream cable. (c) At a large S_R, the controlled tandem cables have two flow patterns. In one type, owing to the vortex pair induced by jet flow, alternating shedding vortices cannot be formed behind the upstream cable. In the other, the upstream cable still forms alternating shedding vortices; however, the vortex pair enlarges the vortex formation length of the upstream cable.

This study verifies that a good control effect can be achieved when both the tandem cables are controlled by passive-suction-jet pipes, which provides a basis for bridge engineering applications of the passive-suction-jet control method.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was funded by the National Natural Science Foundation of China (NSFC) (51722805 and 51978222) and the Fundamental Research Funds for the Central Universities (HIT.BRETIV. 201803).

ORCID iDs

Yan-Jiao Guo

Xiang-Wei Min

References

Alam

Moriya

Takai

, et al (2003) Fluctuating fluid forces acting on two circular cylinders in a tandem arrangement at a subcritical Reynolds number. Journal of Wind Engineering and Industrial Aerodynamics 91(89): 139–154.

Assi

GRS

Bearman

Carmo

, et al. (2013) The role of wake stiffness on the wake-induced vibration of the downstream cylinder of a tandem pair. Journal of Fluid Mechanics 718: 210–245.

Assi

GRS

Bearman

Meneghini

. (2010) On the wake-induced vibration of tandem circular cylinders: The vortex interaction excitation mechanism. Journal of Fluid Mechanics 661: 365–401.

Baek

Karniadakis

(2009) Suppressing vortex-induced vibrations via passive means. Journal of Fluids and Structures 25(5): 848–866.

Chen

, et al. (2020) Suppression of vortex-induced vibration of a circular cylinder by a passive-jet flow control. Journal of Wind Engineering and Industrial Aerodynamics 199: 104119.

Chen

Gao

Yuan

, et al. (2015) Passive jet control of flow around a circular cylinder. Experiments in Fluids 56(11): 201.

Chen

(2014) An experimental study on a suction flow control method to reduce the unsteadiness of the wind loads acting on a circular cylinder. Experiments in Fluids 55(4): 1707.

Chen

Min

Gao

, et al. (2018) Experimental investigation of aerodynamic forces and flow structures of bionic cylinders based on harbor seal vibrissa. Experimental Thermal and Fluid Science 99: 169–180.

Chen

Wang

Gao

(2019) Self-suction and self-jet control on wind loads and turbulent flow structures over a circular cylinder. Journal of Aerospace Engineering 32(1): 04018132.

10.

Dehkordi

Moghaddam

Jafari

(2011) Numerical simulation of flow over two circular cylinders in tandem arrangement. Journal of Hydrodynamics 23(1): 114–126.

11.

Dipankar

Sengupta

Talla

(2007) Suppression of vortex shedding behind a circular cylinder by another control cylinder at low Reynolds numbers. Journal of Fluid Mechanics 573: 171–190.

12.

Feng

Wang

(2010) Circular cylinder vortex-synchronization control with a synthetic jet positioned at the rear stagnation point. Journal of Fluid Mechanics 662: 232–259.

13.

Gao

Chen

, et al. (2017) Flow around a circular cylinder with slit. Experimental Thermal and Fluid Science 82: 287–301.

14.

Jing

(2020) Energy transmission at subcritical Reynolds numbers for the wake-induced vibration of cylinders in a tandem arrangement. Ocean Engineering 211: 107572.

15.

Igarashi

(1981) Characteristics of the flow around two circular cylinders arranged in tandem (1st report). JSME International Journal Series B-fluids and Thermal Engineering 24(188): 323–331.

16.

Kim

Choi

(2005) Distributed forcing of flow over a circular cylinder. Physics of Fluids 17(3): 033103.

17.

Kitagawa

Ohta

(2008) Numerical investigation on flow around circular cylinders in tandem arrangement at a subcritical Reynolds number. Journal of Fluids and Structures 24(5): 680–699.

18.

Lim

Lee

(2002) Flow control of circular cylinders with longitudinal grooved surfaces. AIAA Journal 40(10): 2027–2036.

19.

Lin

Yang

Rockwell

(2002) Flow past two cylinders in tandem: Instantaneous and averaged flow structure. Journal of Fluids and Structures 16(8): 1059–1071.

20.

Ljungkrona

Norberg

Sundén

(1991) Free-stream turbulence and tube spacing effects on surface pressure fluctuations for two tubes in an in-line arrangement. Journal of Fluids and Structures 5(6): 701–727.

21.

Moeller

Leehey

(1984) Unsteady forces on a cylinder in cross flow at subcritical Reynolds numbers. In: Païdoussis

Griffin

Sevik

(eds) ASME Symposium on Flow-Induced Vibrations, vol. 1, New York: ASME, pp.57–71.

22.

Norberg

(2003) Fluctuating lift on a circular cylinder: review and new measurements. Journal of Fluids and Structures 17(1): 57–96.

23.

Owen

Bearman

Szewczyk

(2001) Passive control of VIV with drag reduction. Journal of Fluids and Structures 15: 597–605.

24.

Skop

Balasubramanian

(1997) A new twist on an old model for vortex-excited vibrations. Journal of Fluids and Structures 11(4): 395–412.

25.

Sumner

Price

Paidoussis

(2000) Flow-pattern identification for two staggered circular cylinders in cross-flow. Journal of Fluid Mechanics 411: 263–303.

26.

Tensi

Boue

Paille

, et al. (2002) Modification of the wake behind a circular cylinder by using synthetic jets. Journal of Visualization 5(1): 37–44.

27.

West

Apelt

(1993) Measurements of fluctuating pressures and forces on a circular cylinder in the Reynolds number range 104 to 2.5×105. Journal of Fluids and Structures 7: 227–244.

28.

Wang

(2007) Suppression of the von Kármán vortex street behind a circular cylinder by a travelling wave generated by a flexible surface. Journal of Fluid Mechanics 574: 365–391.

29.

Chen

Xiao

, et al. (2014) Numerical study on the suppression of the vortex-induced vibration of an elastically mounted cylinder by a traveling wave wall. Journal of Fluids and Structures 44: 145–165.

30.

Zhou

(2004) Strouhal numbers in the wake of two inline cylinders. Experiments in Fluids 37(2): 248–256.

31.

Zdravkovich

(1987) The effects of interference between circular cylinders in cross flow. Journal of Fluids and Structures 1(2): 239–261.

32.

Zdravkovich

Pridden

(1977) Interference between two circular cylinders; Series of unexpected discontinuities. Journal of Wind Engineering and Industrial Aerodynamics 2(3): 255–270.

33.

Zhou

Yiu

(2006) Flow structure, momentum and heat transport in a two-tandem-cylinder wake. Journal of Fluid Mechanics 548(-1): 17–48.

34.

Zhu

Yao

, et al. (2015) Simultaneous CFD evaluation of VIV suppression using smaller control cylinders. Journal of Fluids and Structures 57: 66–80.

35.

Zhu

Zhao

Zhou

(2019) Direct numerical simulation of flow over a slotted cylinder at low Reynolds number. Applied Ocean Research 87: 9–25.