Investigation and control of vortex-induced vibration of twin box girders

Journal of Fluids and Structures 39 (2013) 205–221

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Investigation and control of vortex-induced vibration of twin box girders Shujin Laima a, Hui Li a,n, Wenli Chen a, Fengchen Li b a b

School of Civil Engineering, Harbin Institute of Technology, Harbin 150090, China School of Energy Science and Engineering, Harbin Institute of Technology, Harbin 150090, China

a r t i c l e i n f o

abstract

Article history: Received 11 January 2012 Accepted 15 October 2012 Available online 21 February 2013

Stationary and dynamic wind tunnel tests of twin box girders with a space ratio of L/D ¼1.70 have been performed in this study. The vortex-shedding phenomenon under stationary and dynamic conditions and vortex-induced vibration are observed and analyzed. The results indicate that regular vortex shedding occurs only at the trailing edge of the downstream box girder under stationary conditions. Although the strength of the vortex is very weak under stationary conditions, it gives rise to vortex-induced vibrations with a lock-in range of 0.570 r Ur r 0.668 in dynamic testing. The higher harmonics of wind speed around the body is observed and is attributed to nonlinear effects from aerodynamic forces. To further study the flow characteristics around the twin box girder when undergoing vortex-induced vibration, a hybrid method combining experiments with numerical simulations is employed. The pressure distributions, energy transfer between the flow and motion of the body, and evolution of flow patterns over vortex-induced vibration process are analyzed based on the computational results. The results indicate that with an increase in oscillation amplitude, strong vortices form in the gap between the two box girders. These vortices impinge on the windward wall of the downstream box girder and cause the flow to separate and re-attach periodically around the windward corners of the downstream box girder. Based on the analysis of the vortices in the gap, five control measures are used in the wind tunnel test to suppress the vortex-induced vibration of the twin box girders, and the most effective control scheme is obtained. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Twin box girders Vortex-induced vibration Vortex structures Wind tunnel test Hybrid simulation

1. Introduction With the increase in bridge spans, bridges become more flexible and have little damping capability. Therefore, dramatic oscillation of the bridges when subjected to wind is more frequently observed. A twin-separated box steel girder configuration, i.e., the bridge deck is composed of two parallel longitudinal girders with an open space between them, may help to improve the aerodynamic stability of a long-span bridge. The two parallel girders are connected by transverse cross-beams. Currently, several super-long span bridges have chosen this section configuration, for example, the Xihoumen suspension bridge (main span: 1650 m, China), the Hong Kong Stonecutters cable-stayed bridge (main span: 1018 m, China) and the Gwangyang suspension bridge (main span: 1545 m, Korea). Although it is proven that twinseparated box girders have higher critical flutter speed than single box girders (Ge and Xiang, 2008), the flow

n

Corresponding author. E-mail address: [email protected] (H. Li).

0889-9746/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jfluidstructs.2012.10.009

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characteristics around the bridge deck become more complicated due to the effects of the gap between the two separated box girders. It is known that when the vortex-shedding frequency is close to the natural frequency of the body, it can cause vortexinduced resonance. Although vortex-induced vibration (VIV) is a kind of limited amplitude vibration and does not directly cause the collapse of a bridge, it can result in large displacements and discomfort to the drivers. In addition, VIVs commonly occur at low wind speeds, so the occurrence probability of VIV is high, resulting in long-term fatigue damage. At present, the VIV phenomena are observed in real bridges (Frandsen, 2000; Fujino and Yoshitaka, 2002; Larsen et al., 2000). Therefore, investigations on VIV of long-span bridges should be conducted. VIV of circular cylinders has been comprehensively studied. Feng (1968) performed some classical experiments of VIVs of circular cylinders in a wind tunnel. These experiments indicated that two amplitude branches existed with a hysteretic between them, and the jump between two branches is associated with a significant jump of phase angle between pressure at 901 and transverse displacement response. Zdravkovich (1982) showed that oscillation amplitudes are correlated with the timing of vortex shedding using flow visualization. In the lower region of the synchronization range, when the amplitude of oscillation reached a maximum, the vortex was shed on the opposite side. However, in the upper region of synchronization, the timing of the vortex shedding changed suddenly such that the vortex was shed on the same side when the amplitude of oscillation reached a maximum. Williamson and Roshko (1988) indicated that the phase jump in Feng’s experiment is attributed to the vortex-shedding mode switch between 2S mode and 2P mode. Brika and Laneville (1993) showed that the jump from the upper branch to the lower one is accompanied by an instantaneous change from 2S mode to 2P mode using flow visualization. Research of VIVs of cylinders is abundant, and comprehensive reviews of this issue have been performed (Bearman, 1984; Gabbai and Benaroya, 2005; Sarpkaya, 2004; Williamson and Govardhan, 2004). Although the mechanisms of VIVs of circular cylinders are well investigated, there are significant differences in the vortex structures and vortex-induced oscillations between the box girder of a bridge and circular cylinders, which is attributed to different aerodynamic configurations. For the box girder of a bridge, the configuration is more streamlined with a large aspect ratio and wind noses in the windward and leeward regions; therefore, the strength of vortex shedding is much lower than that of a circular cylinder. As many long-span bridges have been constructed, more and more attentions are paid to the VIVs of bridge decks. Diana et al. (2006) investigated the vortex-shedding phenomena of multiple box deck of Messina Strait Bridge and proposed a numerical model to reproduce the vortex shedding forces. Chen et al. (2007) studied the aerodynamic interference between two parallel box-girder bridges. It is concluded that the aerodynamic interference has significant effects on the VIV of two decks. Larsen et al. (2008) investigated the vortex response of a twin box bridge with and without guide vanes. It is observed that the displacement thickness must be at the order of 10% of the guide vane offset in order to allow sufficient flow rate to render the guide vane efficient. They also found that vortices shedding from the upwind box will impinge on the downwind box, resulting in higher fluctuating pressures here than on the upwind box from where the vortices shed. Li and Ge (2008) investigated the effects of size and locations of guide vanes on the response of VIV of a twin-box girder. The results showed that the VIV of a twin-box girder is very sensitively to the size and location of guide vanes. Zhang et al. (2008) studied the VIV of a twin-box girder at low and high Reynolds number. The results indicated that VIV existed in broad range of damping ratio and the amplitude is larger at low Reynolds number. Ge et al. (2011) studied the effects of location of maintenance rail, location of guide vanes, wind barriers, and grid plates installed on the upper and bottom of the gap on VIV of twin-box girders. The results showed that the location of maintenance rail has little influence on the amplitude of VIV. The guide vane installed on the lower surface near the gap, grid plates and wind barriers can effectively suppress the VIV. Although a lot of researches have been conducted, it is still far away to completely understand the mechanism of vortex-induced oscillations of the complex multiple-box girders, in particular the flow characteristics around the multiple-box girders. In addition, the new effective measures to suppress the VIV of the suspension bridge with complex girder shape are also needed to be developed. The main objective of the study is to investigate the flow characteristics around twin box girders and to gain a deeper understanding of the vortex-induced vibration mechanism of twin box girders. The organization of the paper is as follows: in Section 2, experiments of vortex-induced vibrations of twin box girders are described. The vortex-shedding characteristics of the stationary model, the response to vortex-induced vibrations and vortex-shedding characteristics in lock-in range are discussed. In Section 3, a hybrid method combining experiments with numerical simulations is employed to study the vortex-induced vibration of the twin box girders. The computational mesh and simulation parameters are described, and the validation of the hybrid method is also performed. The pressure distribution, energy transfer and flow pattern evolution throughout the vortex-induced vibration process are discussed. In Section 4, five measures are designed to suppress the vortex-induced vibration of twin box girders, and the reduction of vortex-induced vibration is validated through a wind tunnel test. 2. Experimental investigation of vortex-induced vibrations of twin box girders 2.1. Experiment set-up The experiments are conducted in a closed circuit wind tunnel, which has a small rectangle test section of 4 m in width, 3 m in height and 25 m in length and a large rectangle test section of 6 m in width, 3.5 m in height and 50 m in length (see Fig. 1). In the small test section, the maximum wind velocity can be 50 m/s, the turbulence intensity of the free stream

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Fig. 1. Sketch of the facility of wind tunnel combined with water flume at HIT.

Fig. 2. Geometry information of twin box girders. (a) Top view and (b) side view (unit: mm).

can be no stronger than 0.46% and the non-uniformity of free-stream is less than 1%. In this study, all experiments are carried out in the small test section. The model is composed of two parallel box girders with an open space of L¼150 mm. The two parallel girders are connected by transverse cross-beams every 360 mm. The length and height of the model are 1440 mm and 87.55 mm, respectively. Detailed geometry of the section model is shown in Fig. 2, and a 3-D sketch of the test device is shown in Fig. 3. Both stationary and dynamic tests are conducted in this study. In the stationary test, the connected parts (marked ‘‘A’’ in the sketch) are rigid bodies, which can provide adequate vertical stiffness. In the dynamic test, all the rigid connected parts are replaced by 8 springs (marked ‘‘B’’ in the sketch) allowing a vertical degree of freedom with a natural frequency of 4.272 Hz and damping ratio of 0.21%, and a torsion degree with natural frequency of 5.371 Hz and damping ratio of 0.25%. The lower surface of the section model is 1.20 m from the bottom of the wind tunnel. The geometric blockage ratio in the testing section is 2.9%; thus the blockage effects can be ignored. Two one-dimensional hotwire probes (Dantec 55P11) are used in the experiment. One is placed at 1.5 m upstream from the leading edge of the section to measure the oncoming free-stream velocity, while the other probe is moved to different locations near the section model to obtain the vortex-shedding information. Detailed locations are shown in Fig. 4. The sampling frequencies of both hotwire probes were set at 1 kHz. Four accelerometers with a sensitivity of 1 g/mmv are placed at the supporting arms to measure the oscillation information of the section model. Each accelerometer is placed 450 mm from the longitudinal axis of the section model with two on either side. The sampling frequencies are also set at 1 kHz. In the stationary test, the inflow wind velocities are in the range of 2–20 m/s, and the corresponding Reynolds numbers are in the range of 1.2e5 1.2e6 (Re ¼UB/n, in which Re is the Reynolds number, U is the inflow velocity, B is the total width of the model, and n is the kinematic viscosity). In the dynamic test, the inflow wind velocities are in the range of 2.00–6.93 m/s, and the corresponding Reynolds numbers are in the range of 1.2e5  4.158e5. In the experiment, torsional and vertical-torsional VIVs are not observed, so this study mainly focused on the characteristics of vertical VIVs.

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Fig. 3. Sketch of the testing device.

Fig. 4. Locations of the hotwire measured positions.

2.2. Vortex shedding characteristics of a stationary twin box girder Considering that the incline portion of upper surface is very small, and most part of upper surface is flat (see Fig. 2(b)), the flow characteristics should not be changed significantly by the upper surface. For the lower half region, the flow characteristics are influenced dramatically by the geometry of lower surface with large incline regions. Therefore, the flow characteristics on the lower half region are mainly concerned in this study. Fig. 5 shows the power spectra of velocities measured by the one-dimensional hotwires at U ¼3.65 m/s and Re ¼2.19e5. According to the results shown in Fig. 5, for the twin box girder with B/D ¼10.23 and L/D ¼1.70 (B is the width of model, D is the depth of model, and L is the width of the gap), only the velocities at the tail of the downstream box girder exhibit predominant frequencies, indicating that the periodic vortex-shedding phenomena only occur in this region. In addition, the strength of the shedding vortex is very weak. The Strouhal number is a critical parameter for VIVs. The relationship between the vortex-shedding frequency and inflow velocity is shown in Fig. 6. The curve in Fig. 6 indicates that the vortex-shedding frequency follows Strouhal’s law such that the vortex-shedding frequency increases linearly with an increase in inflow velocity. The estimated Strouhal number (St ¼fvD/U, where St is the Strouhal number, and fv is the vortex-shedding frequency) is approximately 0.15 for this type of cross-section. 2.3. Vortex-induced vibration of a twin box girder 2.3.1. Vortex-induced response Although the strength of vortex shedding is weak for the twin box girder under stationary conditions, vertical vortexinduced oscillations occur in the dynamic test. The VIV tests were conducted on the dynamic testing system, as shown in Fig. 3. The vertical natural frequency of the free vibration system is 4.272 Hz, and the corresponding structural damping ratio is 0.21%. The Scruton number, Sc is 59.4 (Sc ¼4pMz/(rairD2), where M is the mass of per length, z is the vertical damping ratio, and rair is the density of air). When carrying out the vertical VIV test, the inflow velocity is increased stepwise, and the increment of each step is approximately 0.05 m/s over the range of 2.19rU r2.65 m/s ð1:30e5 r Re r 1:59e5Þ. Fig. 7 presents the time history of vertical acceleration when the twin box girder is undergoing

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Fig. 5. Power spectra of fluctuating velocity at various locations along the lower-surface of twin box girder for L/D ¼1.70 (inflow comes from the left).

Fig. 6. Variation of vortex-shedding frequency with inflow velocity.

Vertical acceleration (m/s2)

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RMS(y)/D

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L/D=1.70, fvertical=4.272Hz, ξ=0.21%

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0.6 0.7 0.8 0.9 1 1.1 1.2 Reduced wind velocity (Ur=U/fverticalB)

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Fig. 8. Vortex response of the twin separated box girders.

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VIVs. The vertical displacement of oscillation is obtained by integrating the vertical acceleration twice. Fig. 8 shows the relationship between the RMS of displacement response and reduced wind velocity (Ur ¼U/fverticalB, in which Ur is the reduced wind velocity and fvertical is the vertical natural frequency of the model). The response amplitude increases in two steps with the increase in the reduced inflow velocity up to Ur ¼0.650. Over the range of 0.570 rUr r0.603, the amplitude increases extremely slowly, whereas over the range of 0.616 rUr r0.650, the amplitude increases very rapidly. At Ur ¼0.668, the amplitude undergoes a small reduction. When the reduced inflow velocity increases to 0.689, the amplitude of vortex-induced vibration abruptly drops to zero, which means that the vortex exciting force suddenly disappears. In the whole process of vortex-induced vibration, the largest RMS of non-dimensional displacement, RMS(y)/D, is 0.0286 at Ur ¼0.650, which is much smaller than that of rigid cylinders, where the maximum is 0.11 at the same Skop–Griffin parameter of SG ¼1.875 (Sarpkaya, 2004). In the range of 0.570 rUr r0.668, the vortex-shedding frequency is controlled by the oscillation frequency of the body, which is identical to the vertical natural frequency of the system. Beyond this range, the vortex-shedding frequency follows Strouhal’s law. The variation of the dimensionless vortex-shedding frequency with the reduced wind velocity is shown in Fig. 9.

2.3.2. Flow characteristics in the lock-in range When the body is undergoing vortex-induced vibration, the flow characteristics around the body will experience significant changes. Fig. 10 shows the flow velocities and corresponding power spectra density functions at various hotwire measuring points along the lower surface and in the gap at Ur ¼0.650. The detailed locations of measured hot-wire points are presented in Fig. 4. As shown in Fig. 10, the velocities in the gap exhibit strong vortex-shedding characteristics, which do not appear in the stationary condition. This indicates that the shear layers from the upper and lower surfaces of the upstream box girder become unstable and alternately roll up in the gap. The vortices form in the gap and then travel along the surface of the downstream box girder. It is interesting that the velocities at some measured points (such as points 7, 8, 9, 13 and 14) exhibit the higher harmonic phenomenon. This may be attributed to the nonlinear effects of flow. If the oscillation of the body at a natural frequency is regarded as a periodic perturbation to the flow around the body, higher harmonic components must appear in the flow because of the existence of the nonlinear term ui(quj/qxi) in the momentum equation (where u is the velocity of flow). Furthermore, the contribution of higher harmonic components is dependent on the strength of perturbation.

Measured point 14

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Measured point 14 f=4.272Hz 8.545 12.940 17.330

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Fequency (Hz) Fig. 10. Wind velocities and corresponding power spectra density functions at various measured points at Ur ¼0.65 (a) in the gap, and (b) along the lower surface.

According to the analysis above, it can be concluded that the flow pattern changes significantly when the twin separated box girder undergoes vortex-induced vibration. The most obvious change is that strong vortices occur in the gap, which does not appear under stationary conditions. Moreover, due to the interaction between flows and the vibrating body, the strengths of vortices at the tail downstream of the box girder are increased.

3. Numerical investigation of vortex-induced vibrations of a twin box girder 3.1. Computational method In the wind tunnel test, the motion of a structure is easy to measure. However, it is very difficult to accurately measure the fluid structures of a flow field, which usually requires very expensive instrument, e.g., PIV. To obtain characteristics of flow structures when the body is undergoing vortex-induced vibration, a hybrid method that combined experiments with numerical simulations (Li et al., 2010) is employed in this paper. For the problem of flow-induced vibration of a bluff body, the analytical model includes two parts: one is fluid dynamics based on the incompressible Navier–Stokes equations; and the other is the dynamics based on the equation of the motion of the body. The two parts are coupled through the no-slip condition, i.e., the velocities of the fluid and the body are identical at the interface. In traditional computational process, the fluid and structure motion equations are solved simultaneously. For the hybrid method, the motion of the structure is already known, which is first obtained from the wind tunnel test or field monitoring, i.e., the flow velocities on the interface are also known. Therefore, the flow-induced vibration problem becomes a moving boundary problem of the flow. Only fluid dynamic equations need to be solved and the motion of the interface is measured from the wind tunnel test or field monitoring, so it is more accurate than the traditional computational method to obtain the flow characteristics.

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The computational domain is discretized with an unstructured mesh, as shown in Fig. 11. The majority of y þ values, which are nearest to the wall, are in the range of 0.105–1.200 (only 5.67% of y þ values higher than 1.200).The numerical simulations are performed with ANSYS CFX 11 code based on the finite volume method. The large eddy simulation (LES) with a Smagorinsky subgrid scale model is employed in the computational process. The advection term is discretized using the Central Difference Scheme in the spatial domain, and the transient term is discretized by the Second Order Backward 1.11B

1.67B

U∞

upper boundary: symmetric

outlet boundary: Opening pressure

11.42D Inlet boundary: Uniform lower boundary: symmetric

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Euler Scheme in the time domain. The time step is set at 0.001 s, and the convergence criteria is such that if the RMS of normalized residual taken over the whole domain is less than 10  4, the iteration process stops. At the inlet boundary, a steady uniform flow velocity is given. At the outlet boundary, an opening pressure condition is applied, and for the upper and lower surfaces of the computational domain, symmetric conditions are employed. 3.2. Validation of the experimental and numerical hybrid method The equation of motion of a single-degree body with linear springs and damping can be written as: my€ þ cy_ þ ky ¼ F L ðU r ,tÞ,

ð1Þ

where m is the mass of the twin separated box girders model, c is the linear damping coefficient of the free vibration system in stationary air, k is the linear stiffness of the free vibration system in stationary air, FL(Ur,t) is the aerodynamic lift force and y denotes the vertical displacement. In the study, m ¼30 kg, c¼3.38 N s/m, k ¼21 592 N/m. In the wind tunnel test, the vertical acceleration of body is measured. And the corresponding velocity and displacement can be obtained by integrating the vertical acceleration once and twice, respectively. Substituting vertical acceleration, velocity and displacement into Eq. (1), the fluctuating lift force can be calculated, which is shown in Fig. 12. While in computational measured value from wind tunnel test calculated value

Vertical displacement of section model (mm)

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simulation, the aerodynamic force can be obtained by the following equation:   Z  @uj @uk pdjk þ 2m þ nj dS, Fk ¼ @xk @xj S

ð2Þ

in which Fk is the aerodynamic force, p is the pressure, m is the viscosity coefficient, u is the velocity of flow, 2m((quj/qxk)þ (quk/qxj)) represents viscous stresses components of flow and nj denotes the direction vector of the boundary surface of the body, S. The computational lift force is also shown in Fig. 12. Comparison of the lift forces which are obtained from wind tunnel test (Eq. (1)) and hybrid simulation indicates that they agree very well with each other. Additionally, according to Eq. (1) and the aerodynamic lift force FL obtained by the hybrid method, the response of the body can be calculated by the Newmark-b method. Fig. 13 shows the calculated results at Ur ¼0.570, 0.616 and 0.650. It can be found that the calculated results agree well with the results of wind tunnel test, indicating that the experimental and numerical hybrid method can accurately predict the aerodynamic force when the body is undergoing vortex-induced oscillations. In addition, it also implies that flow characteristics are credible. More comprehensive validations of hybrid simulation method have been conducted in the paper (Li et al., 2010). 3.3. Numerical results 3.3.1. Pressure distribution of a twin box girder when undergoing vortex-induced vibrations The pressure coefficient Cp is defined as: pp1 , Cp ¼ 1=2rair U 2

ð3Þ

where p is pressure, pN is the pressure of the free stream and rair is the density of air. Figs. 14 and 15 show the mean and RMS values of the pressure on the surface of the free-vibration model at Ur ¼0.570, 0.616 and 0.650, respectively. The pressure distributions of the stationary model at Re¼1.314e5 are also shown in these figures. For Ur ¼0.570 and 0.616, where the amplitudes of oscillations RMS(y)/D¼0.0049 and 0.0078, the mean pressures are almost the same as in the stationary condition. However, due to the effect of structure oscillation, the RMS pressures are slightly larger than those in the stationary condition, especially on the surface downstream of the box girder. For Ur ¼0.650, where the amplitude of oscillation RMS(y)/D¼0.0286, the pressures change significantly. The pressures present much higher fluctuation characteristics on the surface of the downstream box girder than in the stationary condition, and the mean pressures show high suction values around the windward corners of the downstream box girder. The significant pressure fluctuations are caused by the strong vortices forming in the gap. When these vortices travel to downstream, they will impinge on the windward surface of downstream box girder, and cause the flow to separate and reattach periodically on the lower surface of downstream box girder. This complicated fluid dynamic phenomenon will be discussed in Section 3.3.3 in detail.

Fig. 14. Mean pressure coefficient distribution on the surface of the twin box girder.

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Fig. 15. Distribution of RMS of fluctuating pressure coefficient on the surface of the twin box girder.

It is worth noting that, although the amplitude of oscillation of the twin box girder reaches the largest value at Ur ¼0.650, the mean pressure distributions on the surfaces of the upstream box girder are almost the same as that under stationary conditions, which is attributed to the flows not separating on the upper and lower surfaces of the upstream box girder except in the windward corners. Comparing these three cases with each other, it is found that the RMS of the fluctuating pressure of the downstream box girder increases with the oscillation amplitude of the body. 3.3.2. Energy transfer between flows and the twin box girder In the vortex-induced vibration process, if the work performed by the aerodynamic force on the body is positive, i.e., the energy of flows is transferred to body, the vibration of the body will gradually increase. The energy transfer is mainly influenced by the amplitude of the fluctuating force and the phase between the fluctuating force and body motion. The aerodynamic force is calculated by integrating the surface pressure of the twin separated box girders. Lift force coefficient per length is defined as: CL ¼

FL ð1=2Þrair U 2 BL

,

ð4Þ

where CL is the lift force coefficient, FL is the lift force, U is the inflow velocity, B is the width of model and L is the length of the model. The phase between the lift force and vertical displacement is calculated by the following equation: 9 8 realðPly ðoÞÞ > > > if imagðP ly Þ Z0 > > > arc cosqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > > > > = < ðreaðPly ðoÞÞÞ2 þ ðimagðP ly ðoÞÞÞ2 , ð5Þ yðoÞ ¼ realðPly ðoÞÞ > > > > > arc cosqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi if imagðP ly Þ o0 > > > > > > > ; : ðrealðPly ðoÞÞÞ2 þ ðimagðP ly ðoÞÞÞ2 where y(o) is the phase lag between lift force and vertical displacement, Ply is the cross power spectrum of lift force and vertical displacement and o is the circular frequency of vertical vibration (o is equal to the oscillation frequency). The total work of lift force performed on the twin box girder is obtained as follows: Z 1 yðKT þ NTÞ F L dy, ð6Þ W¼ N yðKTÞ where W is the work of lift force on the body, T is the period of vertical vibration and K and N are the numbers of cycles. Fig. 16 shows the mean lift force coefficient, RMS of fluctuating lift force coefficient, phase angle between lift force and vertical displacement and work of lift force acting on the twin box girder at Ur ¼0.570, 0.583, 0.603, 0.616 and 0.650. At Ur ¼0.570, 0.583 and 0.603, the phase between lift force to vertical displacement are 72.01,  95.01 and  110.01,

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0.1

-0.01

RMS(CL)

Mean(CL)

0

-0.02 -0.03 -0.04 0.5

0.55

0.6

0.65

0.05

0 0.5

0.7

0.6

0.65

0.7

0.55

0.6

0.65

0.7

x 10-4

Work of lift force on body (J)

-60

Phase (Degree)

0.55

-80 -100 -120 -140 -160 0.5

0.55

0.6

0.65

0.7

Reduced wind velocity (Ur=U/fverticalB)

6 4 2 0 0.5

Reduced wind velocity (Ur=U/fverticalB)

Fig. 16. Lift force characteristics and work performed by the lift force at Ur ¼ 0.570, 0.583, 0.603, 0.616 and 0.650. (a) Mean lift force coefficients, (b) RMS of fluctuating lift force coefficient, (c) phase between lift force and displacement, and (d) work performed by the lift force .

respectively, which are nearly in phase with the vertical velocity of body vibration, indicating that the aerodynamic lift force mainly provides negative damping characteristics. However, the net work acting on the structure is very low, which is attributed to a small-fluctuation lift force, so the amplitude of vibration is small at the beginning of oscillation. At Ur ¼0.650, although the phase of the lift force to vertical displacement is about 144.01, which deviates from the direction of the vertical velocity, the large-fluctuation lift force resulted in large work of lift force on the body, so a dramatic response of the body is observed. It is found that the phase lags between lift force and vertical displacement are very different from those for a circular cylinder, in which phase lags are less than 45.01 in initial branch of oscillation, and lead to an increase in oscillation (Feng, 1968). Unlike a circular cylinder, no phase jump over the vortex-induced vibration process appears for the box girder. Substituting Eq. (2) into Eq. (6), the work of lift force on the body can be rewritten as:   Z Z  @uj @uk 1 yðKT þ NTÞ Wk ¼ pdjk þ 2m þ nj dSdy N yðKTÞ @xk @xj S   Z yðKT þ NTÞ  Z @uj @uk 1 pdjk þ 2m þ ð7Þ nj dydS: ¼ @xk @xj S N yðKTÞ To obtain the energy transfer over the whole surface of the body, a work density function P(x,y,z) on the surface of the twin box girder is defined as:    Z @uj @uk 1 yðKT þ NTÞ pdjk þ2m þ ð8Þ nj dy: Pðx,y,zÞ ¼ N yðKTÞ @xk @xj The work density of flow over the surface of the twin box girder at Ur ¼0.570 and Ur ¼0.650 at z¼630 mm is shown in Fig. 17. The positive values of the work density show that the flow energies are transferred to body, while the negative values indicate that the energy of body motion is transferred to the flows. At Ur ¼0.570, the beginning of vortex-induced vibration, the energy of flows is transferred to the body in the regions of X/B ¼[0.217 0.412] for the lower surface and X/B ¼[0.190 0.450] for the upper surface due to the influence of shedding vortices in the near wake. However, for the case of Ur ¼0.650 (the amplitude of response is largest), besides the work density becoming much larger, the region where the energy of the flows transfers to the body is located mainly near the gap. This implies that the vortices in the gap play a critical role to vortex-induced vibrations of twin box girders. 3.3.3. Flow pattern evolution when the twin box girder undergoes vortex-induced vibration In the vortex-induced vibration process, the structure motion and vortex interact with each other, forming a feedback loop system. Therefore, the vortex structure and vortex-shedding modes have significant effects on the structural oscillation. Lugt (1979) gave a classical definition of a vortex structure, that is ‘‘A vortex is the rotating motion of a magnitude of material particles around a common center’’. This intuitive definition describes well the observation of vortices in nature. However, it ¨ is not Galilean invariant under moving reference frames, so it is not valid in general (Rutten et al., 2008). Thus far, the most widely used local vortex definition criteria are based on properties of the velocity gradient tensor rV, for example, Q-criterion (Hunt et al., 1988), D-criterion (Chong et al., 1990; Dallmann, 1983), l2-criterion (Jeong and Hussain, 1995), etc. Although the vortex structures can be recognized according to these criteria, there still lacks a common agreement, and these methods

S. Laima et al. / Journal of Fluids and Structures 39 (2013) 205–221

Work density , P(x,y.z) (J/m2)

8

217

x 10-4 lower surface upper surface

6

U

4 2 0 -2 -4 -6 -0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

X/B

Work density , P(x,y.z) (J/m2)

0.03

lower surface upper surface

0.02 U

0.01 0 -0.01 -0.02 -0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

X/B Fig. 17. Work density over the surface of the twin box girder at z ¼630 mm (a) Ur ¼ 0.570, and (b) Ur ¼0.650.

have their respective advantages and limitations. In this study, the l2-criterion is selected to obtain the vortex structure. Jeong and Hussain (1995) argued that, if discarding unsteady straining and viscous effects, the existence of a minimum pressure value and existence of a vortex core are consistent. By taking the gradient of the Navier–Stokes equation and decomposing it into symmetric and asymmetric parts, the following equation can be deduced: DSij 1 vSij,kk þ Oik Okj þ Sik Skj ¼  p,ij , Dt r

ð9Þ

where Sij is the strain tensor, Sij ¼ (1/2)((qui/qxj)þ(quj/qxi)), Oij is the rotation tensor, Oij ¼(1/2)((qui/qxj)  (quj/qxi)) and p,ij is the Hessian pressure. The occurrence of a local pressure minimum in a plane requires two positive eigenvalues of the tensor p,ij. Discarding the first term of Eq. (9), which represents an unsteady irrotational straining effect and the second term, which represents viscous effects, yields only S2 þ O2 to determine the existence of a local pressure minimum. Therefore, the vortex core definition criterion is that the vortex is the location with two negative eigenvalues of S2 þ O2. Assuming l1, l2 and l3 are the eigenvalues of S2 þ O2 and l1 r l2 r l3, the vortex identification criterion is equivalent to the condition that l2 o0. Fig. 18 presents the vortex core structures at Ur ¼0.570 and Ur ¼0.650. At the initial stage of vortex-induced vibration, i.e., Ur ¼0.570, the body motion leads to the boundary layers of the upstream box girder becoming unstable and generates vortices on the upper and lower sides of the gap. It is worth noting that these vortices are generated due to the flow by the body motion disturbance rather than the direct interaction of the upper and lower shear layers. The shedding vortices propagate downstream along the upper and lower surfaces of the downstream box girder and then merge with the vortices generated in the inclined regions of the downstream box girder. Finally, the merging vortices are shed alternately in the near wake of the downstream box girder with the 2S shedding mode, i.e., when the displacement is largest, the single vortex sheds from the opposite side. Due to the asymmetrical geometry of the wind nose (inclined regions of box girder), asymmetrical vortex structures are generated in the upper and lower sides of the near wake. For the lower side of the nose, the width of the inclined plane is very long (Linclined/B ¼0.180), so the nascent vortex on the lower side of the wake is long and narrow, and the energy distribution of the vortex is not uniform. This special kind of vortex structure results in instability of the primary vortex on the upside of wake and rapid translation into 3-D structures. However, for the upper side of the nose, the dimensionless width of the inclined plane Linclined/B is only 0.056, so the primary vortex is more regular and stays stable when propagating downstream. When the amplitude of the vortex-induced vibration is largest, the vortex-shedding phenomena and vortex structures change significantly. The interaction between the upper and lower boundary shear layers of the upstream box girder

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Fig. 18. Vortex core structures, left: iso-surfaces of l2 ¼  10 s  2; right: the contour plots of l2 at section of Z ¼ 630 mm (a) Ur ¼0.570 and (b) Ur ¼ 0.650.

is enhanced through the gap, and thus, strong vortices generate around the gap. These vortices propagate downstream along the surfaces of the downstream box girder and finally break into small vortices in the nose region. In the near wake of the downstream box girder, because of body motion disturbance, the primary vortices, which appear at the initial stage of vortex-induced vibrations, disappear, and the flow transforms into turbulent flow. As shown in Fig. 19, the vortexshedding phenomena around the gap are very complicated. As the body reaches its maximum displacement, the nascent vortex of the side impinges on the windward wall of downstream box girder and cuts off the vorticity supply of the opposite vortex, so a free vortex forms. In addition, the flow from the opposite side separates around the windward corner of the downstream box girder and a new vortex begins to form at that moment. After approximately 1/4 of a cycle, i.e., the displacement of body is zero, the separated flow begins to re-attach, and the flow re-attaches on the surface of the downstream box girder completely as the body reaches the maximum displacement on the same side.

4. Control of vortex-induced vibrations of twin separated box girders The above analysis indicates that strong vortices are generated in the gap when the response amplitude of vortexinduced vibrations is large. Therefore, if the vortices in the gap are disturbed, the vortex-induced oscillation may be suppressed. Based on this knowledge, five schemes are proposed to control the vortex-induced vibration and the control effectiveness is investigated through the wind tunnel test. The schemes are presented as follows: Scheme 1, cross bars are installed on the upper and lower sides of the gap, as shown in Fig. 20(a); Scheme 2, cross bars are installed on the upper side of the gap only, as shown in Fig. 20(b); Scheme 3, cross bars are installed on the lower side of the gap only, as shown in Fig. 20(c); Scheme 4, cross bars are installed in the gap, as shown in Fig. 20(d); and Scheme 5, the gap is covered using light flat plates, as shown in Fig. 20(e). Fig. 21 shows the amplitude of vibration of the girder under these five schemes. For Schemes 1 and 5, the oscillation amplitudes of the girder are close to zero, so the vortex-induced vibrations are completely suppressed. For scheme 2, after the critical point Ur ¼0.62, the vortex-induced vibration is completely suppressed, while before the critical point, the vortex-induced vibration increases a little. For scheme 3, the control effectiveness is almost the same as Scheme 2 except that the amplitude of the vortex-induced vibration does not increase before the critical point. For scheme 4, the maximum amplitude only decreases by 25.5% and the amplitude increases approximately 30.0% before the critical point. It must be noted that the main purpose of adopting twin-box girder is to improve the flutter stability. Any countermeasures of suppressing vortex-induced vibrations of bridge deck should not reduce the flutter critical wind speed. Scheme 5 that covering the gap will weaken the flutter stability of bridge deck. Fortunately, the vortex-induced vibrations of long-span suspension bridge occur in low wind speed range (Li et al., 2011), while the critical flutter wind speed is very high. Therefore, for practical application, Scheme 5 should be designed as automatically covering and opening the gap according to the feedback of the measured wind speed and vibration of the bridge deck, which are readily measured by the implemented structural health monitoring system.

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219

Fig. 19. Instantaneous vorticity contours in a periodic vibration at z ¼630 mm.

5. Conclusion The experimental investigations of vortex-induced vibrations of a twin box girder model with a gap ratio L/D ¼1.70 are carried out in this study. The following conclusions can be obtained: Under stationary conditions, there is no occurrence of regular Karman vortices in the gap, but regular vortex-shedding phenomena with St ¼0.15 appear in the near wake of the downstream box girder. Although the strength of the primary vortex is very weak in the stationary test, vortex-induced vibration of the twin box girder occurs in the dynamic test. The synchronization range is 0.570 rUr r0.668. Furthermore, the largest RMS of non-dimensional displacement is only 0.0286 at Ur ¼0.650, which is much smaller than that of circular cylinders. Due to the perturbation caused by body oscillation, the fluid near the gap presents the higher harmonic phenomenon. A hybrid method is employed to investigate the flow characteristics around the twin box girders when undergoing vortex-induced vibrations. The results show that the regular vortices are generated only in the near wake of the downstream box girder with the 2S shedding mode at the initial stage of vortex-induced vibration. However, as the amplitude of oscillation becomes large, strong vortices are generated in the gap and the flow of near wake transforms into turbulent flow. The vortices impinge on the windward wall of the downstream box girder when propagating downstream and cause the flow to separate and re-attach periodically around the windward corners of the downstream box girder. Due to the effects of vortices in the gap, the pressure shows high fluctuating characteristics near the windward corner of the downstream box girder. The energy of body oscillation is generated mainly by the pressures of these regions. Five schemes are designed to suppress the vortex-induced vibration of twin box girders. The results show that a plate covering the gap can achieve the best suppression of vortex-induced vibration. From a practical standpoint, automatically opening or covering the gap is the most convenient and effective approach in applying this method to real bridges.

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Scheme 1

Scheme 2

Scheme 3

Scheme 4

Scheme 5

Fig. 20. Five different controlling schemes to suppress the vortex-induced vibration: (a) cross bars on the upper and lower side of the gap, (b) cross bars on the upper side of the gap, (c) cross bars on the lower side of the gap, (d) cross bars in the gap, and (e) light flat plates on the upper and lower side of the gap.

RMS(y)/D

0.03 uncontrolled scheme 1 scheme 2 scheme 3 scheme 4 scheme 5

0.02

0.01

0 0.4

0.5

0.6

0.7

0.8

0.9

1

Reduced wind velocity (Ur=U/fverticalB) Fig. 21. Relationship between RMS of vertical displacement and reduced wind velocity under five different controlling schemes.

Acknowledgments This study is financially supported by the NSFC under Grant nos. 90815022 and 51161120359. References Bearman, P.W., 1984. Vortex shedding from oscillating bluff bodies. Annual Review of Fluid Mechanics 16, 195–222. Brika, D., Laneville, A., 1993. Vortex-induced vibration of a long flexible circular cylinder. Journal of Fluid Mechanics 250, 481–508. Chen, Z.Q., Niu, H.W., Li, C.G., 2007. Experimental study on wind-induced vortex shedding of parallel box-girder bridge. Journal of Hunan University (Natural Sciences) 34 (9), 16–20. (in Chinese). Chong, M.S., Perry, A.E., Cantwell, B.J., 1990. A general classification of three-dimensional flow fields. Physics of Fluids A 2 (5), 261–265. ¨ Dallmann, U., 1983. Topological Structures of Three-dimensional Flow Separations. DFVLR. Rep. No. 221-82-A07. Gottingen, West Germany. Diana, G., Resta, F., Belloli, M., Rocchi, D., 2006. On the vortex shedding forcing on suspension bridge deck. Journal of Wind Engineering and Industrial Aerodynamics 94, 341–363. Feng, C.C., 1968. The Measurement of Vortex-induced Effects in Flow Past Stationary and Oscillating Circular and d-Section Cylinders. Master’s Thesis. Department of Mechanical Engineering, The University of British Columbia, Canada. Frandsen, J.B., 2000. Simultaneous pressures and accelerations measured full-scale on the Great Belt East suspension bridge. Journal of Wind Engineering and Industrial Aerodynamics 89, 95–129. Fujino, Y., Yoshitaka, Y., 2002. Wind-induced vibration and control of Trans-Tokyo Bay Crossing Bridge. Journal of Structural Engineering 128 (8), 1012–1025. Gabbai, R.D., Benaroya, H., 2005. An overview of modeling and experiments of vortex-induced vibration of circular cylinders. Journal of Sound and Vibration 282, 575–616. Ge, Y.J., Xiang, H.F., 2008. Bluff body aerodynamics application in challenging bridge span length. In: Proceedings of 6th International Colloquium on Bluff Bodies Aerodynamics and Applications. July 20–24, Milano, Italy.

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